Galois Groups of Periodic Points

201, 401]428 Ž1998. JA977304

JOURNAL OF ALGEBRA ARTICLE NO.

Galois Groups of Periodic Points Patrick Morton Department of Mathematics, Wellesley College, Wellesley, Massachusetts 02181 Communicated by Walter Feit Received April 1, 1997

1. INTRODUCTION In this paper we take another look at the Galois groups of the polynomials Fn Ž x . s

Ł Ž f dŽ x. y x. d
m Ž nrd .

,

where f is a polynomial with coefficients in a field K, f d is the dth iterate of f with itself, and m Ž d . is the Moebius function. The roots of FnŽ x . are periodic points of the map f in an algebraic closure of K: generally, such a root has n as its minimal period, but some roots can have smaller period Žcf. w11x, w12x.. These Galois groups have been studied previously w19, 1, 11, 17, 7x. In several of these papers the emphasis was on special complex maps: in w1x Bousch computed these groups for the quadratic polynomial f Ž x . s x 2 q c over K s CŽ c .; Lau and Schleicher w17, 7x Žamong other things. extended Bousch’s result to the maps f Ž x . s x k q c, for k G 2. These results imply that for f Ž x . s x k q c, the Galois group of FnŽ x . over CŽ c . is the wreath product of ZrnZ with a symmetric group S r Žcrf. Theorem 4.2 of w11x., i.e., a ‘‘generalized symmetric group’’ in the language of w5x. The computations given by both Bousch and Lau]Schleicher contain an essentially analyticalrtopological aspect, since they use paths on Riemann surfaces to exhibit specific automorphisms, as well as detailed knowledge of the structure of the Mandelbrot set Žand its analogue for k ) 2.. It is of interest to know whether an algebraic derivation can be given for these and related Galois groups. If so, it might be possible to compute the Galois group of FnŽ x . over fields of positive characteristic or over algebraic number fields. 401 0021-8693r98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.

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PATRICK MORTON

In this paper I show that such an approach is possible. The methods used here extend the methods in w9x and w11x Žincluding ideas from w8x and w13x. and are applicable in fairly generally situations. In w9x we showed that FnŽ x . is absolutely irreducible for certain families of polynomials f in two variables x and c, either over Q Žor C. or over Fp Žwith additional restrictions on f and p .. In this paper we first assume irreducibility of FnŽ x . over a more general field and give conditions on its discriminant which are sufficient to imply that its Galois group is the generalized symmetric group mentioned above. We then show how the results of Bousch, Lau, and Schleicher can be deduced, and prove some analogous results for more general polynomials over arbitrary fields. To state our main result, assume that f Ž x . g Rw x x is monic, where R is a Dedekind domain with fraction field K. As is shown in w13x, the discriminant of FnŽ x . has the form disc Fn Ž x . s "Dnn , n ?

Ł

d < n , dFn

d Dny n, d ,

Ž 1.

for certain elements D n, d of R Žsee Section 2.. The factors D n, d determine the collapse or coincidence of periodic n-orbits of the map f Žmod p ., for prime ideals p of R Žsee w13x.. For instance, a prime ideal p of R divides D n, 1 if and only if some root of FnŽ x . is actually a fixed point of f Žmod p ., and p divides D n, n if and only if two of the orbits which make up the roots of FnŽ x . coincide Žmod p .. These facts are closely related to the way the prime factors of disc FnŽ x . ramify in the field K Ž a . generated over K by a root of FnŽ x .. We prove the following result. MAIN THEOREM. Let f Ž x . g Rw x x be monic in x, where R is a Dedekind domain with a quotient field K, and let S be the splitting field of f Ž x . o¨ er K. Assume that Ži. FnŽ x . is irreducible o¨ er K; Žii. Some prime ideal p di¨ ides D n, 1 to the first power but is relati¨ ely prime to D n, d for all d / 1 and d < n; Žiii. Ž D n, n . / 1 is a square-free ideal in R which is relati¨ ely prime to D n, d for all d / n and d < n. Let a be a root of FnŽ x . in a splitting field S of FnŽ x . o¨ er K and let nr s deg FnŽ x .. Then either Ža. there is a proper, nontri¨ ial subextension L of K in K Ž a . which is unramified o¨ er all prime ideals of R and fixed by the automorphism a ª f Ž a .; then GalŽ FnŽ x .rK . s GalŽ SrK . ( ZrnZ wr G, where G is an imprimiti¨ e subgroup of Sr ; or Žb. the Galois group GalŽ FnŽ x .rK . s GalŽ SrK . ( ZrnZ wr Sr . This theorem follows by an argument whose main ingredient is the ramification theory for finite extensions of Dedekind domains. In this

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GALOIS GROUPS OF PERIODIC POINTS

respect the results of this paper have a certain resemblance to the results of w16x, in which Odoni gives algebraic proofs, using ramification theory, of his earlier, analytically derived, results w15x. If R is the ring of integers in an algebraic number field K with no nontrivial unramified extensions, such as K s Q, K s QŽ i ., or K s QŽ62., then conclusion Ža. is impossible, so that conditions Ži. ] Žiii. always imply Žb. over K. In other words, if FnŽ x . is irreducible over K, information about its discriminant is sometimes sufficient to determine its Galois group completely! Conclusion Ža. is also impossible if n F 6 and the narrow class number of K is relatively prime to 6. I conjecture that Ža. is impossible over the ring of integers of any algebraic number field K whose narrow class number is 1. In both cases of the Main Theorem GalŽ SrK . ( ZrnZ wr GalŽ LrK ., for a certain subfield L of S; if F is the fixed field inside K Ž a . of the automorphism Ž a ª f Ž a .., then L is the normal closure of F over K. Thus GalŽ LrK . s G and Sr , respectively, in cases Ža. and Žb.. This fact allows us to exhibit Dedekind domains R over which case Ža. of the Main Theorem occurs. For example, consider a quadratic f Ž x . and F5 Ž x . satisfying the hypotheses of the Main Theorem for n s 5 over Z. There are r s 6 orbits of 5-periodic points of f ; let G be the largest imprimitive subgroup of S6 which permutes the blocks  1, 2, 34 ,  4, 5, 64 . Then G has order 72, so the fixed field K of G inside L has degree 10 over Q. It is not hard to see that F5 Ž x . is still irreducible over K and Žsince K is a subfield of L . that GalŽ SrK . s ZrnZ wr G. If S denotes the set of prime ideals of K which have ramification index 2 over Q and R S denotes the ring of S-integers in K Želements of K whose denominators are only divisible by primes in S ., then R S is a Dedekind domain and conditions Žii. and Žiii. hold for the polynomial F5 Ž x . and its discriminant factors D 5, 1 and D 5, 5 over R S Žsee Example 1 in Section 3.. The following result gives an indication of how special the extensions are which occur in the Main Theorem. THEOREM A. Assume that conditions Ži. and Žiii. of the Main Theorem hold and that the class number of the domain R is odd. Let F be the fixed field of the automorphism a ª f Ž a . inside K Ž a . and let L be the normal closure of F o¨ er K. Then, for some unit « in R, the extension LrK Ž « D n , n . is unramified o¨ er all prime ideals of the integral closure of R in K Ž « D n , n ..

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The corollary to the Main Theorem in Section 3 shows that the field L is actually unramified over many of its subfields, in case Žb.; this stems from the fact that the inertia groups in GalŽ LrK . are all generated by transpositions. In this case GalŽ LrK Ž « D n , n .. s A r Žthe alternating group on r letters .. Thus all instances of the Main Theorem over the ring R s Z

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PATRICK MORTON

give rise to unramified A r extensions of the quadratic fields QŽ « D n , n .. An interesting special case occurs for f Ž x . s x 2 q c and n s 5, when r s 6.

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COROLLARY TO THEOREM A. If the polynomial F5 Ž x . corresponding to the map f Ž x . s x 2 q c, for some c g Z, is irreducible o¨ er Q, and if the integer D 5, 5 Ž c . s 4,194,304c 11 q 32,505,856c 10 q 109,051,904c 9 q 223,084,544c 8 q 336,658,432 c7 q 402,464,768c 6 q 379,029,504c 5 q 299,949,056c 4 q 211,327,744c 3 q 120,117,312 c 2 q 62,799,428c q 28,629,151 is square-free and relati¨ ely prime to D 5, 1Ž c . s 256c 4 q 64 c 3 q 16 c 2 y 36 c q 31, then the field L of Theorem A is an unramified extension of QŽ y D 5, 5 . with Galois group A 6 .

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Using this corollary with c s y1 and c s y4, respectively, we find that the fields QŽ'y 7 = 527053 . and QŽ'1,665,040,211,057 . Žthe latter has prime discriminant. both have unramified A 6 extensions which are subfields of the fields generated by the fifth order periodic points of x 2 y 1 and x 2 y 4, respectively. The Main Theorem also applies over function fields K. Case Ža. turns out to be impossible if K s k Ž c ., where k is a perfect field, if FnŽ x . is absolutely irreducible over k and if the ramification at infinity Žat the pole divisor p` of c in K . is assumed to be tame. This allows us to prove the following result, of which the BouschrLau]Schleicher results are special cases Žsee Section 4.. For its statement we require the following definition. If representatives of the r orbits of roots of FnŽ x . under the map f are a i , 1 F i F r, then the multiplier of the ith orbit is v i s Ž f n .X Ž a i . s dŽ f n Ž x ..rdx < xs a i and the nth multiplier polynomial is r

dn Ž x . s

Ł Ž x y vi . .

is1

If the coefficients of f lie in the ring R, then so do the coefficients of dnŽ x . w19, 11x and dnŽ1. is related to the factors D n, d of disc FnŽ x . by the formula dnŽ1. s D n, n Ł d < n, d - n D n, d . With this definition we may state THEOREM B. Let k be an arbitrary field and let f Ž x, c . g k w x, c x be a polynomial of degree k G 2 in x, satisfying the following conditions: Ži. Žii. Žiii. Živ.

for some m G 1, f Ž x, u m . is homogeneous in x and u; f Ž x, 0. s x k ; f Ž x, 1. has distinct roots; dnŽ1. s dnŽ1, c . has distinct roots.

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GALOIS GROUPS OF PERIODIC POINTS

Then FnŽ x . and dnŽ x . are irreducible o¨ er k with Galois groups Gal Ž Fn Ž x . rk Ž c . . ( ZrnZ wr Sr , Gal Ž dn Ž x . rk Ž c . . ( Sr , where nr s deg FnŽ x .. ŽThese isomorphisms are also ¨ alid o¨ er k Ž c ... The genus of the curves FnŽ x . s FnŽ x, c . s 0 and dnŽ x . s dnŽ x, c . s 0 in this theorem can be found in w9x. By Theorems 11 and 13 of w9, Sect. 3x the genus results in w9, Theorem Cx hold under the more general hypotheses of Theorem B given here, whenever the field k is a perfect field. These ideas also lead to the following results. THEOREM C. Let k be any perfect field, and assume that f Ž x, c . g k w x, c x satisfies conditions Ži. ] Žiii. of Theorem B. If the polynomial DnŽ c . s Ł d < n d d Ž1, c . has distinct roots, then the following isomorphism holds for the Galois group of f n Ž x . y x o¨ er k Ž c .: Gal Ž f n Ž x . y xrk Ž c . . s

m Gal Ž F Ž x . rk Ž c . . ( m ZrdZ d
d

d
wr Sr d ,

with drd s deg Fd Ž x .. In other words, periodic points of different periods generate independent splitting fields. THEOREM D Žsee Section 4, Theorem 10.. If f Ž x . s x k q c, with k G 2, then the Galois group of f n Ž x . y x o¨ er QŽ c . is the direct product of the Galois groups of its irreducible factors: Gal Ž f n Ž x . y xrQ Ž c . . s

m Gal Ž F Ž x . rQŽ c . . ( m ZrdZ d
d

d
wr S r d ,

where drd s deg Fd Ž x .. Furthermore, if PerŽ f . denotes the set of all periodic points of f in an algebraic closure of QŽ c ., then the Galois group of the field generated by PerŽ f . is the unrestricted direct product of wreath products: Gal Ž Q Ž Per Ž f . . rQ Ž c . . s

`

Ł Ž ZrdZ wr Sr . s Wk , d

ds1

where r d s Ž1rd .Ý e < d m Ž dre. k e. If MultŽ f . denotes the set of multipliers of all orbits of f in PerŽ f ., then GalŽQŽMultŽ f ..rQŽ c .. s Ł`ds 1 Sr d and QŽPerŽ f .. is an abelian extension of QŽMultŽ f .. with Galois group Gal Ž Q Ž Per Ž f . . rQ Ž Mult Ž f . . . (

`

Ł Ž ZrdZ.

rd

.

ds1

Finally, QŽMultŽ f .. is unramified o¨ er the compositum Ł`ns 1QŽ D n , n Ž c . . at finite primes.

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PATRICK MORTON

This theorem shows that QŽPerŽ f .. is an abelian extension of an unramified extension of a compositum of hyperelliptic function fields. Of course the isomorphisms in Theorem D, which are implicit in the results of w1x, are possible to prove only because so much is known about the parameter spaces of the complex maps f Ž x . s x k q c. This result depends heavily on the way that the closures of the hyperbolic components in these parameter spaces intersect each other Žsee w3x, w4x, w17x and the proof of Theorem 9.. For any fixed polynomial f Ž x . in Qw x x of degree k, the Galois group of the field generated over Q by its periodic points injects into the unrestricted direct product Wk . It would be interesting to know, and is probably very hard to determine, whether GalŽQŽPerŽ f ..rQ. is equal to Wk for any single polynomial f, say f Ž x . s x 2 q 2 Žsee Example 2 in Section 3.. 2. RAMIFICATION THEORY OF FnŽ x . We begin by reviewing the setup from w11x: f Ž x . is a monic polynomial over a field K and FnŽ x . is the polynomial defined in the Introduction. We will assume that FnŽ x . is irreducible over K. If a i is a root of FnŽ x . in a splitting field S of FnŽ x . over K, then we set K i s K Ž ai . and let Fi denote the fixed field in K i of the automorphism Ž a i ª f Ž a i ... In the case being considered here, that FnŽ x . is irreducible, the extension K irFi is a cyclic extension of degree n with generating automorphism Ž a i ª f Ž a i .. w11, Corollary to Theorem 4.5x. The minimal polynomial of a i over Fi is ny1

li Ž x . s

Ł Ž x y f j Ž ai . . ; js0

hence w Fi : K x s deg FnŽ x .rn s r, the number of orbits of f which make up the roots of FnŽ x .. If, furthermore, the a i , for i s 1, . . . , r, are roots taken from each of these r orbits, then the fields F1 , . . . , Fr are conjugate to each other in S and their compositum L is normal over K. The fields Fi correspond 1]1 to the orbits A i s  a i , f Ž a i ., . . . , f ny 1 Ž a i .4 of roots of FnŽ x ., and the invariant group N of L inside G s GalŽ SrK . is the kernel of the map which associates to an automorphism s in G the corresponding permutation of the orbits A i . Thus GrN s GalŽ LrK . is naturally a permutation group on these orbits and on the fields Fi Žor on the polynomials l i Ž x ... The elements of the group N may be represented as

GALOIS GROUPS OF PERIODIC POINTS

407

r-tuples

s s Ž f s1 , . . . , f s r . in which the ith coordinate f S i represents the effect of s on the orbit A i : s Ž a i . s f s i Ž a i .. Thus we have injections GrN ª Sr and N ª Ž ZrnZ . r w11, Theorem 4.1x. Theorem 4.2 of w11x also gives the injection G s Gal Ž SrK . ª Ž ZrnZ .

wr Ž GrN . .

Under certain conditions this injection is an isomorphism. From w11x we have LEMMA 1.

If < N < s n r , then GalŽ SrK . ( Ž ZrnZ . wr Ž GrN ..

We will also make use of the discriminant theory in w13x. There it is proved that the formula Ž1. holds, where D n, d is defined as follows. If dnŽ x . s Ł is1, . . . , r Ž x y v i ., where v i is the multiplier of the orbit A i , and d d Ž x . denotes the corresponding polynomial for period d, then D n , d s Res Ž Cn r d Ž x . , d d Ž x . . , for d < n, d - n, where Cn r d Ž x . is the nrdth cyclotomic polynomial, and

dn Ž 1 . s D n , n

Ł

d < n , d-n

Dn, d .

Ž 2.

Furthermore, Res Ž Fn Ž x . , Fd Ž x . . s "Ddn , d , if d < n, d - n,

Ž 3.

so that D n, d s 0 if and only if some orbit A i is really a d-orbit repeated nrd times, i.e., if and only if some orbit of roots of FnŽ x . ‘‘collapses.’’ If the coefficients of f lie in an integral domain R, then the D n, d lie in R as well. We will later consider the case in which K s k Ž c . and R s k w c x, where k is a field, but it is convenient to consider a more general situation. For the rest of this section we assume that R is a Dedekind domain with quotient field K, f Ž x . g Rw x x is monic in x, K i is the field K Ž a i ., RX is the integral closure of R in K i . Ž RX is then also a Dedekind ring.. The next two propositions give information about the ramification of the primes p of R in K irK. We will either use the term ‘‘prime ideal’’ or ‘‘prime divisor’’ when speaking of the primes of R and its extensions.

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PATRICK MORTON

PROPOSITION 2. Ža. If a prime ideal p of R is ramified in LrK Ž or in Fi ., then p di¨ ides D n, n . Žb. If a prime di¨ isor ` of p in RX l Fi is ramified in K irFi , then p di¨ ides D n, d , for some proper di¨ isor d of n. Proof. Ža. Assume that ` is a prime divisor of L for which ` 2 < p and let P be a prime divisor of S lying above `. The inertia group GT of ` in GalŽ LrK . is nontrivial, so there exists an automorphism s / 1 in GT for which s Ž a . ' a Žmod `. for all a in RY , the integral closure of R in L. Since s is not 1, there are i and j with i / j for which s Ž l i Ž x .. s l j Ž x ., and therefore l j Ž x . ' l i Ž x . Žmod `.. It follows that l j Ž a i . ' l i Ž a i . ' 0 Žmod P ., hence P divides D n, n by the formula D n , n s h2 Ł l j Ž a i . , i/j

which is Eq. Ž1.9. in w13x Žh 2 is a unit in the integral closure of R in S .. This implies that p < D n, n as well. Žb. The minimal polynomial of a i over Fi divides l i Ž x .. By the same argument as in part Ža. we see that f i Ž a i . ' a i Žmod P . for some j, 0 - j - n, where P is a prime divisor of ` in K i . Let d - n be the smallest positive integer with the property that f d Ž a i . ' a i Žmod P .. Then d < n, Fd Ž a i . ' 0 Žmod P . and p divides Ž D n, d . d s "ResŽ FnŽ x ., Fd Ž x .. from Ž3.. This proves Žb.. Proposition 2 holds even when FnŽ x . is reducible, since no assumption is made about the size of the groups GalŽ K irFi .. This is not the case in the following proposition, which relies on the fact that GalŽ K irFi . is cyclic of order n. PROPOSITION 3. Assume that FnŽ x . is irreducible in K w x x and that the prime ideal p of R di¨ ides dnŽ1. to exactly the first power. Ži. The prime di¨ isors P of RX which di¨ ide p are in 1]1 correspondence with the irreducible factors of FnŽ x . Žmod p .. Žii. If p di¨ ides D n, d , where d < n and d - n, then p is unramified in FirK; exactly one of its prime di¨ isors ` in Fi ramifies in the cyclic extension K irFi , and then with ramification index nrd. The relati¨ e degree of ` o¨ er p is 1. Žiii. If p is a prime di¨ isor of D n, n , then p factors in Fi as p s ` 12 ` 2 ??? ` k with exactly one ramified prime di¨ isor Ž` 1 .; the relati¨ e degree of ` 1 o¨ er p is 1. The ` j are all unramified in K irFi .

GALOIS GROUPS OF PERIODIC POINTS

409

Proof. Ži. By w6, p. 279x the prime divisors P of RX which divide a given prime divisor p of R are in 1]1 correspondence with the irreducible factors of FnŽ x . over the field K Ž p ., the completion of K with respect to p. We need to show that different irreducible factors g and h give rise to relatively prime polynomials in the residue field Rrp. Assume therefore that g and h are factors of FnŽ x . in K Ž p .w x x which have the common irreducible factor uŽ x . Žmod p .. Let P and Q be the prime divisors of RX corresponding to g and h. Then uŽ x . is of course a multiple factor of FnŽ x . Žmod p ., so that p divides the discriminant of FnŽ x ., and a i Žmod P . and a i Žmod Q . are roots of uŽ x . in RXrP and RXrQ, respectively. Using that a i is a double root of FnŽ x . Žmod P . and Žmod Q . it follows from the formula 1 y vi s

d dx

Ž x y f n Ž x . . < xs a s yFnX Ž a i . Ł Fd Ž a i . i

d < n , d/n

Ž 4.

that P and Q both divide 1 y v i . Hence the prime divisors ` 1 and ` 2 of Fi lying below P and Q also divide Ž1 y v i .. Now the norm to K of Ž1 y v i . divides dnŽ1., and the norms of ` 1 and ` 2 to K are both equal to a power of p, so ` 1 must equal ` 2 with relative degree 1 over p. Thus P and Q lie over the same prime divisor ` 1 of Fi and are therefore conjugate by some power of the automorphism s s Ž a i ª f Ž a i ... Since the minimal polynomial of a i over Fi is l i Ž x ., P and Q correspond to irreducible factors g X and hX of l i Ž x . over the ` 1-adic completion of Fi . Since the residue class field of Fi mod ` 1 equals Rrp, and uŽ x . is the minimal polynomial of a i Žmod P . and a i Žmod Q . over Rrp, it follows that uŽ x . is a factor of both g X and hX mod ` 1 and hence a multiple factor of l i Ž x .. Hence two of the roots of l i Ž x . must be congruent mod P: let d - n be the smallest positive integer with the property that f d Ž a i . s a i Žmod P .. As in the proof of Proposition 2Žb., d < n, Fd Ž a i . ' 0 Žmod P . and p divides D n, d . Now we recall the formulas FnX Ž a i . s lXi Ž a i . Ł l j Ž a i . ,

Ž 5.

j/i

lXi Ž a i . s h

Ł

d < n , d-n

Fd Ž a i .

Ž nrdy1 .

,

Ž 6.

r where h is a unit in RX ; the first follows from FnŽ x . s Ł is1 l i Ž x . and the second is w9, Eq. Ž20.x Žsee also the proof of Theorem 2.5 in w13x.. Formulas Ž4. ] Ž6. show that P n r d divides 1 y v i , and since 1 y v i has a square-free divisor in Fi , the ramification index eŽ P <` 1 . of P over ` 1 is at least nrd Hence the inertia group GT of P over ` 1 has order G nrd. On the other hand, none of the powers s j, for 1 F j F d y 1, lies in GT , since a i k

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PATRICK MORTON

f j Ž a i . Žmod P .. Thus the powers s j, for 0 F j F d y 1, lie in distinct cosets of GT in GalŽ K irFi .; consequently < GT < s nrd. Now l i Ž x . has exactly d distinct roots mod ` 1 , and its irreducible factors Žmod ` 1 . all have the same degree Žbecause their roots comprise a periodic orbit of the map f ; see w11, Theorem 4.5Žb.x.. It follows that ` 1 has at least drd distinct prime divisors in K i , where d s deg uŽ x ., each one with degree G d over ` 1 Žand hence over p .. This implies that P has degree d and drd distinct conjugates over Fi , each one corresponding to a distinct factor of l i Ž x . Žmod ` 1 .. Hence, g X s hX and P s Q, completing the proof of Ži.. Now assume p is a prime divisor of D n, d , for d < n, d - n. Then by Ž3., FnŽ x . and Fd Ž x . have a common factor over Rrp which corresponds to a unique prime divisor P of K i . Since P divides Fd Ž a i ., a i has period d with respect to f Žmod P .; it doesn’t have smaller period since p cannot divide any of the terms D n, dX with dX / d, by hypothesis. Hence the preceding argument applies and P has ramification index nrd over ` 1 Žin the notation of the preceding paragraph.. Now the power of p which divides the discriminant of K irK is at least

Ž Norm K

P Ž n r dy1. .

dr d

s p Ž n r dy1. d s p nyd ,

which is exactly the power of p that divides disc FnŽ x .. Since discŽ K irK . divides discŽ FnŽ x .. it follows from Ž1. that p ny d is the exact contribution of p to discŽ K irK .. Hence p has exactly one prime divisor in Fi Žof degree 1. which ramifies in K i , and p is unramified in FirK, by Proposition 2Ža., since p does not divide D n, n . This proves Žii.. Now suppose that p divides D n, n . By Ž2. and the hypothesis p does not divide D n, d for any d / n; thus P is not ramified over Fi by Proposition 2Žb.. If P divides the prime divisor ` of Fi and ` e exactly divides p, then p lŽ ey1. would divide disc FirK Ž p l s Norm K `. and p lŽ ey1. n would divide disc K irK, by the Schachtelungssatz w6, pp. 424 and 448]449x. However, Ž1. shows that exactly the nth power of p divides discŽ FnŽ x .., which implies that l s 1 and e F 2. Hence at most one prime divisor of p in Fi can be ramified over p, and in that case can only have relative degree 1. It remains to see that for some prime `, ` 2 exactly divides p. Because p < D n, n the polynomial FnŽ x . has a multiple root Žmod p .. Thus there is an irreducible factor uŽ x . of FnŽ x . and a prime divisor P of p in K i for which P < uŽ a i .. In particular, uŽ x . divides both l i Ž x . and FnŽ x .rl i Ž x . Žmod p ., since l i Ž x . has no multiple roots Žmod p .. Formulas Ž4. and Ž5. show that P divides 1 y v i , so that any prime divisor P X of S lying above P also divides 1 y v i ? . Factoring FnŽ x .rl i Ž x . s Ł j/ i l j Ž x . modulo P X shows that there is a j / i for which uŽ x .< l j Ž x . Žmod P X . Žsince the roots of uŽ x . form an orbit under some iterate of f .. But then P X < uŽ a j ., for the appropriate choice of a j , whence P X < l i Ž a j . and P X <Ž1 y v j ., by Ž4. and Ž5..

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It follows that Ž P X . 2 divides dnŽ1., and so Ž P X . 2 divides p. Thus p ramifies in S. The above argument now shows that p must ramify in Fi , as claimed. This proves Žiii. and also that the exact contribution of p to disc K irK is p n. Q.E.D. Remark. Under the hypothesis of Proposition 3 the ramification over p is always tame, as the above proof shows. When K is an algebraic number field, it follows that if p divides nrd Žfor d - n. and p divides D n, d , then p 2 < dnŽ1.. Also, if p is a prime divisor of 2 and p divides D n, n , then p 2 < dnŽ1.. COROLLARY 1. Assume FnŽ x . is irreducible in K w x x, and that the ideal Ž dnŽ1.. is square-free in R. Ži. The discriminant of K irK equals the di¨ isor Ždisc FnŽ x ... Žii. The ring RX s Rw a i x has the R-integral basis  1, a , . . . , a dy 14 , where a s a i and d s deg x FnŽ x .. Proof. Since Ž dnŽ1.. is square-free, Proposition 3 applies to all of the primes p dividing dnŽ1., which are all the primes that can possibly ramify in K i , by Ž1. and Ž2.. The proof of the proposition shows that discŽ K irK . and discŽ FnŽ x .. are equal as divisors, and this implies that the powers of a i are an integral basis for RX over R. COROLLARY 2. Assume FnŽ x . is irreducible in K w x x, and that the ideal Ž D n, n . is square-free in R and relati¨ ely prime to Ž D n, d . for all d < n, d - n. Ži. The discriminant of FirK equals Ž D n, n .. Žii. Assume in addition that R has odd class number. Then, for a suitable unit « of R the extension LrK Ž « D n , n . is unramified o¨ er all prime ideals of the integral closure RY of R in K Ž « D n , n ..

'

'

Proof. Since the only possible prime divisors of the discriminant of FirK are prime divisors of D n, n , and D n, n is square-free, Ži. follows from the proof of part Žiii. of Proposition 3. To prove Živ., we use the fact that the discriminant Ž D n, n . of FirK is a principal ideal. It follows that a generator g of Fi which is integral over R has discriminant Ž dŽg .. s Ž D n, n . I 2 , for some ideal I of R. Since R has odd class number and Ž dŽg .. is principal the ideal I must also be principal. Hence dŽg . s « D n, n ? s 2 , for some unit « and some element s of R, so that K Ž « D n , n . is certainly a subfield of the normal closure L. By the tameness of the ramification in FirK and part Žiii. of Proposition 3 it follows that the ramification indices of ramified primes in LrK are all 2 Žsee the statement of Abhyankar’s lemma given below.. But this is also true of the ramification indices in K Ž « D n , n .rK, and since the same primes ramify in L and in K Ž « D n , n ., L must be unramified over the prime ideals of RY . Q.E.D.

'

'

'

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Note that Corollary 2Žii. coincides with Theorem A of the Introduction. Proposition 3 is fundamental in all that follows. In particular, it allows us to show that N has order n r. To prove this we first recall the following form of Abhyankar’s lemma w18, p. 125x. ŽThis statement is a slight modification of the statement in w18x, but the proof is exactly the same.. ABHYANKAR’S LEMMA. Let R be a Dedekind domain with quotient field K,and let K X be a finite extension of K, where K X s K 1 K 2 is the compositum of two intermediate fields K : K 1 , K 2 : K X . Let P be a prime di¨ isor of R, let P X be one of its extensions to K X , and let Pi s the restriction of P X to K i . If at least one of the extensions PirP is tame, then the ramification index of P XrP is related to the ramification indices of the Pi by e Ž P XrP . s lcm eŽ P1rP ., eŽ P2rP .4 . PROPOSITION 4. Let f Ž x . g Rw x x be monic in x. Assume that FnŽ x . is irreducible o¨ er K and that there is a prime di¨ isor p of K which di¨ ides D n, 1 to the first power but does not di¨ ide Ł d < n, d / 1 D n, d . Then < N < s w S:L x s n r. Proof. I claim first that w K i L:L x s n for each i s 1, . . . , r. Certainly w K i L:L x F w K i : Fi x s n. To show this degree equals n, note there is a prime divisor ` of p in Fi which ramifies totally in K i , by Proposition 3Žii.. Suppose P is a prime divisor of ` in the composite extension K i L. The ramification index of P in K i LrFi is a multiple of n. Because p is unramified in F1 F2 ??? Fr s L, it follows that the ramification index of P in K i LrL is at least n, whence w K i L:L x G n also. This proves the claim. Hence we also get that K i l L s Fi . Now we show that the fields K i L are linearly disjoint over L. To do this we compute the contribution of p to the discriminant of K i L, which we denote simply by d i s p-discŽ K i L .. Let ` be the prime of Fi dividing p which ramifies in K i . The relative degree and ramification index of ` over p are both 1, so ` splits in L into a product of l primes of relative degree f and ramification index 1 over p, where lf s w L: Fi x. Therefore all the primes dividing p in L have degree f s w L: Fi xrl, and p is a product of w L: K xrf s w L: Fi xw Fi : K xrf s rl prime divisors ` 1 , ` 2 , . . . , ` r l . Let P X be a prime divisor in K i L of some ` i and let P and `X be the primes of K i and Fi which P X divides. By Abhyankar’s lemma, the ramification index of P X over `X , which on the one hand is eŽ P X <`X . s eŽ P X <` i . ? eŽ` i <`X . s eŽ P X <` i ., is given by e Ž P X <`X . s lcm  e Ž P <`X . , e Ž ` i <`X . 4 s lcm  e Ž P <`X . , 1 4 s e Ž P <`X . , and therefore eŽ P X <` i . s eŽ P <`X .. Since ` is the only prime divisor of p which ramifies in K irFi , exactly l of the prime divisors ` i ramify totally in

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413

K i LrL, namely, those which divide `, and the others are unramified, giving us that di s Ž ` 1 ` 2 , . . . , ` l .

ny 1

,

with a suitable numbering. ŽIf the ramification is not tame, then d i s ` 1e1 ` 2e 2 ??? ` le l , with e i G n, but this doesn’t affect the following argument.. Now the fields K i L are all conjugate under the action of G s GalŽ SrK ., so the divisors d i are also conjugate. Furthermore, the primes ` j are all conjugate to each other under G, so every ` j Ž1 F j F rl . divides some d i . Since each of the r divisors d i has l distinct prime divisors and p has rl prime divisors in all Žin L ., no prime ` j can appear in more than one d i , and these discriminants are pairwise relatively prime. Thus the fields K i L, for i s 1, . . . , r, have disjoint ramification over p and are consequently linearly disjoint over L Žsince the ramification over p in K i LrL is total ramification.. In other words, r

w S:L x s w K 1 K 2 ??? K r :L x s Ł w K i L :L x s n r .

Q.E.D.

is1

The argument of Proposition 4 is very similar to the arguments of Lemmas 2 and 3 in w8x. COROLLARY. Assume that FnŽ x . is irreducible o¨ er K and that for some prime di¨ isor p of K, p ny 1 exactly di¨ ides discŽ FnŽ x ... Then < N < s w S:L x s n r and GalŽ SrK . ( Ž ZrnZ . wr GalŽ LrK .. Proof. The hypothesis on p, together with formula Ž1., implies that p divides D n, 1 , but does not divide Ł d < n, d / 1 D n, d . To see this, we just have to show that p cannot divide a combination of terms D n, d for which S d Ž n y d . e d s n y 1, for some nonnegative integers e d . Certainly p does not divide D n, n and does not divide a lone term D n, d , with d / 1 or n, since the exponent n y d of D n, d in Ž1. is divisible by d. If p divides at least two such terms, then the exponent of p dividing discŽ FnŽ x .. would be at least 2Ž n y nr2. s n. This proves the assertion. The conclusion now follows from Proposition 4 and Lemma 1. It seems quite remarkable that a simple divisibility condition for the discriminant of FnŽ x . implies a structural isomorphism for its Galois group. There does not exist a similar criterion for prime divisors p of D n, d for general d Ž/ n., as Example 2 in w11, Sect. 6x shows: if f Ž x . s x 2 y x q 1 over K s Q and n s 4, then p s 3 divides D 4, 2 to the first power and divides neither of D 4, 1 and D 4, 4 , but N has index 2 in the full group Ž Zr4Z . 3.

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In the situation of the last corollary, the permutation Ž a ª f Ž a .. on the roots of FnŽ x . coincides with the automorphism Ž f, f, . . . , f ., which definitely lies in N since N ( Ž ZrnZ . r. Thus the corollary gives a condition, valid for general n, under which the permutation represented by the map f is an automorphism in GalŽ SrK .. In w11x this was expressed by saying that f is an automorphism polynomial for S. Žsee w11, Sect. 7x..

3. PROOF OF THE MAIN THEOREM Under fairly general hypotheses we can also prove certain facts about the group GalŽ LrK .. PROPOSITION 5. Assume that FnŽ x . is irreducible o¨ er K and that for some prime di¨ isor p of K, p di¨ ides D n, n to exactly the first power but does not di¨ ide D n, d for any other d / n. Then, as a permutation group on the fields Fi Ž or on the orbits A i ., GalŽ LrK . contains a transposition. Proof. The assertion of Proposition 3Žiii. is valid for the prime p. Let g Ž x . be the irreducible polynomial in Rw x x Žof degree r . satisfied by some primitive element of FirK. The prime divisors of p in Fi are then 1]1 correspondence with the irreducible factors of g Ž x . over K Ž p ., the completion of K with respect to p, in such a way that, if the prime ` corresponds to the factor hŽ x ., then deg hŽ x . s ef, where e and f are the ramification index and relative degree of ` over p, respectively. Hence g Ž x . has a quadratic factor h1Ž x . corresponding to the prime ` 1 whose square divides p, by Proposition 3Žiii.. The field F Ž`. generated by a root of h1Ž x . is a totally ramified quadratic extension of K Ž p . contained in the completion LŽ P . of L with respect to some prime P which divides ` 1 in L. Note that LŽ P . is a splitting field for g Ž x . over K Ž p ., so the other factors of g Ž x . over K Ž p . have roots that generate unramified extensions L j of K Ž p . inside LŽ P .. The compositum LX of the L j is unramified over K Ž p . and even normal over K Ž p . since it is clearly mapped to itself by any automorphism of LŽ P .rK Ž p .. Therefore, F Ž`. l LX s K Ž p .. It follows that there is an autormorphism s of LŽ P .rK Ž p . which is the identity on LX but has order 2 on F Ž`., and s may be viewed as an automorphism in GalŽ LrK . by the standard inclusion GalŽ LŽ P .rK Ž p .. ª GalŽ LrK .. As a permutation on the roots of g Ž x ., s is a transposition. This proves the proposition. In the next proposition we give a condition for GalŽ LrK . to be a primitive permutation group on the orbits.

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PROPOSITION 6. Assume that FnŽ x . is irreducible o¨ er K, and that Ž D n, n . is a square-free ideal in R which is relati¨ ely prime to D n, d for all proper di¨ isors d of n. Then either Ži. the group G s GalŽ LrK . is a primiti¨ e permutation group on the set of fields Fi or Žii. the field Fi contains a nontri¨ ial extension of K which is unramified o¨ er all primes of R, and G is imprimiti¨ e. Proof. The permutation group G is transitive on the fields Fi , so G is primitive if and only if point stabilizers are maximal subgroups of G w21x. The stabilizer of Fi is just Gi s GalŽ LrFi ., and Gi is maximal in G if an only if FirK has no intermediate field. Suppose there is such a field L, K ; L ; Fi . By Proposition 2Ža. the only primes of R which could possibly ramify in LrK are divisors of D n, n , and by Corollary 2 to Proposition 3, discŽ FirK . s Ž D n, n .. If p divides D n, n , I claim p is not ramified in LrK either. Suppose p does ramify in L. Then the square of some prime ideal ` 2 divides p in L. If ` ramifies in FirL, then some prime divisor of p in Fi would have ramification index ) 2 over p, which is impossible by Proposition 3Žiii.. If ` is not ramified in FirL, then either ` splits in Fi into more than one prime ideal, in which case p would have two ramified prime divisors in Fi , or ` remains inert in Fi , in which case a ramified prime divisor of p in Fi has relative degree ) 1 over p. Both of these situations are impossible, by Proposition 3. This shows that all prime divisors p of D n, n are unramified in L, so L is a nontrivial, unramified extension of K. Q.E.D. Putting Propositions 4, 5, and 6 together leads to the Main Theorem Žstated in the Introduction.. Proof of the Main Theorem. Certainly GalŽ SrK . ( ZrnZ wr G holds in either case, with G s GalŽ LrK ., by Lemma 1 and Proposition 4. By Proposition 6 either G is primitive or there is a subfield L of Fi satisfying the assertion of Ža.. In either case, Proposition 5 implies that G : Sr contains a transposition, since D n, n must have at least one prime divisor, by hypothesis. If G is primitive, then because any primitive subgroup of Sr containing a transposition must be equal to Sr Žsee w21, Theorem 13.3, p.34x., we get the assertion of Žb.. Q.E.D. Remark. If K is a field which has no unramified extensions, such as K s Q, then conclusion Ža. is impossible and we can dispense with the hypothesis that Ž D n, n . / 1; in such a situation Ž D n, n . necessarily has at least one prime divisor, as long as r / 1. If r s 1, then GalŽ SrK . ( ZrnZ, as is implied in both cases Ža. and Žb.; this occurs, for example, when n s 2 and f is quadratic.

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EXAMPLE 1. Let f Ž x . s x 2 y 3 and n s 5. Then F5 Ž x . is irreducible over Q with discriminant disc F5 Ž x . s 1014 = 1914 = 2815 = 75915 = 10,243 5, so that D 5, 1 s "101 = 191 and

D 5, 5 s "281 = 7591 = 10,243.

Hence the conditions of the Main Theorem are satisfied over R s Z and GalŽ F5 Ž x .rQ. s ŽZr5Z. wr S6 . Now let G be the largest imprimitive subgroup of S6 which permutes the blocks  1, 2, 34 ,  4, 5, 64 , as in the Introduction. Then G has order 72, so the fixed field K of G inside L has degree 10 over Q. If a i is a root of F5 Ž x ., then wQŽ a i . L:L x s 5, by the proof of Proposition 4, and it follows that QŽ a i . l L s Fi . Moreover, Fi corresponds Žwlog. to the subgroup H of S6 fixing the digit 1, so an easy Galois theoretic argument shows that Fi l K s Q. This implies that F5 Ž x . is irreducible over K. Furthermore, since K is a subfield of L, neither of the primes 101 or 191 ramify in K, so Ž D 5, 1 . is a square-free ideal in the ring of integers R of K and Proposition 4 and Lemma 1 give GalŽ SrK . s ZrnZ wr G. Thus, conditions Ži. and Žii. of the Main Theorem hold over R. Condition Žiii. does not hold over R, because the primes dividing D 5, 5 must ramify in K. However, there are prime divisors of K which divide D 5, 5 to just the first power. To see this, recall that the ramification indices in LrQ of prime divisors P of D 5, 5 are 2. Hence the inertia group GT Ž P . s  1, t 4 , for an involution t . Any such involution has conjugates which lie in G and conjugates which do not. Suppose therefore that t does lie in G. Then GT Ž P . l G s  1, t 4 is the inertia group of P over K and P lies over some prime ` of K all of whose prime divisors in L have ramification index 2. It follows that the ramification index of ` in KrQ is 1, and therefore ` divides D 5, 5 to exactly the first power. Now let S be the set of prime ideals of K which have ramification index 2 over Q and let R S be the subring of K which consists of the S-integers in K. Then R S is a Dedekind domain and conditions Žii. and Žiii. hold for the polynomial F5 Ž x . and its discriminant factors Ž D 5, 1 . and Ž D 5, 5 . over R S . This is because the prime ideals of R S are in 1]1 correspondence with the prime ideals of R not lying in S Žcf. w14, pp. 71]75x., and prime ideal factorizations of elements of R S are gotten from those in R by just suppressing the occurrence of primes in S. This shows that case Ža. of the Main Theorem holds for the map f Ž x . and the polynomial F5 Ž x . over the Dedekind domain R S . When applied to a primiti¨ e subgroup of S r , the argument used in Example 1 leads to the following fact, for which we give a more straightforward proof.

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COROLLARY TO THE MAIN THEOREM. Assume that conditions Ži. ] Žiii. and case Žb. of the Main Theorem hold for the map f and polynomial FnŽ x .. Let LrK be any nontri¨ ial subextension of LrK with the property that GalŽ LrL. is primiti¨ e, as a subgroup of Sr . Then the extension LrL is unramified at all prime ideals of RŽ L., the integral closure of R in L. Proof. Let p be any prime ideal of R which divides Ž D n, n . and let P be one of its prime ideal factors in L. With notation as in the proof of Proposition 5, the completion LŽ P . is a ramified quadratic extension X Ž p . is unramified. It follows that the inertia group of the field LX , and LrK of P in GalŽ LrK . is GT Ž P . s GalŽ LŽ P .rLX . s  1, s 4 , where s is a transposition, when considered as a permutation on the r orbits of n-periodic points. If L is a nontrivial extension of K contained in L and G s GalŽ LrL. is primitive, then the transposition s cannot lie in G, by w21, Theorem 13.3 p. 34x. Hence the inertia group of P over L is trivial and P is not ramified over L. In particular, the group A r is a primitive subgroup of Sr , so this corollary gives a second proof of Theorem A in case Žb.. EXAMPLE 2. Let f Ž x . s x 2 q 2 over R s Z. If n s 4, then F4 Ž x . s x 12 q 12 x 10 q x 9 q 66 x 8 q 8 x 7 q 209 x 6 q 28 x 5 q 404 x 4 q 49 x 3 q 454 x 2 q 40 x q 241. This polynomial is irreducible over Q with discŽ F4 Ž x .. s 13 2 = 53 3 = 1439 4 . This D 4, 1 s 53, D 4, 2 s y13, and D 4, 4 s 1439 are all prime, and GalŽ F4Ž x .rQ. s ŽZr4Z. wr S3 , a group of order 4 3 = 6 s 384. It is interesting to factor F4Ž x . modulo the primes 13, 53, and 1439, in order to verify the assertions of Proposition 3. We have F4 Ž x . ' Ž x q 1 . Ž x q 2 . Ž x q 7 . Ž x q 10 . Ž x 2 q x q 3 . = Ž x 4 q 4 x 3 q x 2 q 2 x q 6.

2

Ž mod 13 . ,

so that 13 has a single ramified prime divisor in K irK with ramification index 4r2 s 2 and degree 2 over p s 13. The fact that the roots of an f-orbit must generate the same field extension over H Zr13Z implies that the roots of x 2 q x q 3 belong to a single f-orbit Žmod 13., illustrating the collapse of the ‘‘ramifying’’ orbit. Also 4

F4 Ž x . ' Ž x q 38 . Ž x 4 q 24 x 3 q 45 x 2 q 49 x q 6 . = Ž x 4 q 36 x 3 q 28 x 2 q x q 42 .

Ž mod 53 . ,

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PATRICK MORTON

agreeing with the fact that 53 divides D 4, 1 , and 2

2

2

F4 Ž x . ' Ž x q 97 . Ž x q 424 . Ž x q 649 . Ž x q 662 .

2

= Ž x 4 q 653 x 3 q 563 x 2 q 1213 x q 1258 .

Ž mod 1439. ,

which shows that 1439 does indeed have a single ramified prime divisor ` in Fi with ramification index 2 and degree 1 over 1439, and that ` even splits completely in K i . In fact, Fn, f Ž x . satisfies the conditions of the Main Theorem for all n F 6 except n s 2 Žwhen r s 1., as we can see in Table I. Irreducibility of each Fn, f Ž x . was checked on Mathematica. ŽThe values of D n, d were computed directly from the polynomials in w13, Table 1x with c s 2..

4. THE FUNCTION FIELD CASE We will apply the Main Theorem in the case that R s k w x x, where k is either an algebraic number field or a finite field. In order to exclude conclusion Ža. of the Main Theorem, we prove the following lemma. LEMMA 7. Let K s k Ž c . be a rational function field in one ¨ ariable, where k is perfect, and let p be a prime di¨ isor of K of degree 1. Then there does not exist a nontri¨ ial, finite, separable extension of K, unramified o¨ er primes / p and tamely ramified o¨ er p, whose field of constants is also k . Proof. Assume that L is a finite separable extension of K, ramified only over p, with field of constants k . Then LX , the normal closure of L over K, is also ramified only over p; suppose k X is its field of constants. X Consider the extension Lr k X Ž c .. Only the prime divisors of p in k X Ž c . can ramify in this extension, and there is exactly one such prime divisor pX over p Žsince the residue class degree of any such pX is w k X : k x.. By The TABLE I Galois Groups of Fn, f Ž x . for f Ž x . s x 2 q 2 n

D n, 1

1 2 3 4 5 6

11 79 53 11 = 421 43

D n, d Ž d < n.

D n, n

GalŽ SrQ.

y13 Ž d s 2.

y7 y1 3=5 1439 7 = 34,703,374,561 y619 = 3797 = 34,862,835,782,647

S2 Zr2 Z Ž Zr3Z . wr S2 Ž Zr4Z . wr S3 Ž Zr5Z . wr S6 Ž Zr6Z . wr S9

157 Ž d s 2. 1249 Ž d s 3.

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Riemann]Hurwitz genus formula the genus g of LX is given by X 2 g y 2 s y2 LX : k X Ž c . q deg Different Ž Lr kX Ž c. . .

Since LX is normal over k X Ž c ., pX splits into w LX : k X Ž c .xrfe prime divisors in LX , each with ramification index e and relative degree f. Hence deg DifferX entŽ Lr k X Ž c .. s Ž e y 1.w LX : k X Ž c .xre Žsince the ramification is assumed to be tame.. It follows that LX : k X Ž c .

Ž y2 q Ž e y 1. re . G y2,

X

L : k X Ž c . F2 er Ž e q 1 . - 2. Hence LX s k X Ž c . and L is contained in the constants extension k X Ž c . of k Ž c .. Since the field of constants of L is also k it follows that L s k Ž c . s K, proving the assertion. If char k s 0 no wild ramification is possible, so the field K s k Ž c . has no extensions Žother than constants extensions . which are ramified over a single prime divisor of K Žof degree 1.. This is all we need in order to apply the Main Theorem for K, since the only prime divisor of K which does not correspond to a prime ideal of R s k w c x is the infinite prime p` , the pole divisor of c. If char k ) 0 we must add an extra hypothesis that the ramification over p` is tame. We obtain the following result. THEOREM 8 Žfor K s k Ž c ... is a perfect field. Assume that

Let f Ž x . g k w x, c x be monic in x, where k

Ži. FnŽ x . is irreducible o¨ er k , the algebraic closure of k ; Žii. D n, 1 s D n, 1Ž c . has at least one simple irreducible factor in k w c x which is relati¨ ely prime to D n, d for all d / 1; Žiii. D n, n s D n, nŽ c . is square-free in k w c x and relati¨ ely prime to D n, d for all d / n; Živ. Ž and when char k ) 0. the pole di¨ isor p` of c is either unramified or tamely ramified in FirK. Then GalŽ Lrk Ž c .. s Sr and GalŽ FnŽ x .rk Ž c .. s GalŽ Srk Ž c .. ( ZrnZ wr Sr , where nr s deg FnŽ x .. Proof. Note first of all that Žii. and Žiii. imply that discŽ FnŽ x .. / 0, so FnŽ x . is a separable polynomial over K. Furthermore, the field of constants of K i is k by hypothesis Ži.. By Proposition 6, either GalŽ LrK . is a primitive permutation group or Fi contains a nontrivial extension L of K s k Ž c . which is ramified only over p` . Hypothesis Živ. implies that L would have to be tamely ramified at p` , so Lemma 7 shows that no such extension exists. This also implies that D n, nŽ c . is nonconstant, since Fi

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would be unramified over K in that case. The conclusion now follows as in the Main Theorem. This theorem applies to all the polynomials which satisfy hypothesis ŽH. in w9x. We now show how the results of w9x can be extended to obtain Theorem B Žstated in the Introduction.. Proof of Theorem B. Case 1. Suppose k is an algebraic number field. Let p be a rational prime which splits completely in k and which has no prime ideal factors in common with m, the denominators of the coefficients of f, the leading coefficient of dnŽ1, c ., or with discŽ f Ž x, 1.. ? discŽ dnŽ1, c .., and let ` be a prime ideal factor of p in k . Then f Ž x, c . ' f 0 Ž x, c .Žmod `., where f 0 is a polynomial with coefficients in Fp s ZrpZ which also satisfies conditions Ži. ] Živ. Žover Fp .. For convenience we may assume that f 0 has coefficients in Z. We will show that Fn, f 0Ž x, c . is absolutely irreducible over Fp ; it will then follow that Fn, f Ž x, c . is absolutely irreducible over k , and we will be able to apply Theorem 8. To show that Fn, f 0Ž x, c . is irreducible Žmod p . we find a polynomial f 1Ž x, c . in Zw x, c x for which f 0 ' f 1 Žmod p ., where f 1 satisfies Ži. ] Živ. and the additional condition that gcdŽdisc f 1Ž x, 1., k n y 1. s 1. Taken together, these conditions are condition ŽH. in w9x, and Theorem 15 of that paper will imply that Fn, f 1Ž x, c . ' Fn, f 0Ž x, c . Žmod p . is absolutely irreducible over Fp . We seek a polynomial f 1Ž x, 1. s f 0 Ž x, 1. q pg Ž x . satisfying Ži., Žii. and gcdŽdisc f 1Ž x, 1., k n y 1. s 1, where deg g F k y 1. The coefficients of f 0 Ž x, 1 . q pg Ž x . s x k q Ž b1 q pa1 . x ky 1 q ??? q Ž bky 1 q paky1 . x q bk q pak are linear in the coefficients a1 , . . . , a k of g Ž x .. For each prime divisor q / p of k n y 1 choose values aŽi q. of the coefficients a i in Fq for which the corresponding polynomial f 0 Ž x, 1. q pg q Ž x . is a given irreducible polynomial in Fq w x x and whose discriminant is therefore nonzero Žmod q .. By the Chinese remainder theorem we can determine integer values of the a i for which a i ' aŽi q. Žmod q . for all primes dividing k n y 1 and with these coefficients it is easy to see that f 2 Ž x . s f 0 Ž x, 1. q pg Ž x . satisfies gcdŽdisc f 2 Ž x ., k n y 1. s 1. If m s 1 in Ži. we set f 1Ž x, c . s c k f 2 Ž xrc .. If m / 1, then write f 0 Ž x, 1. as hŽ x m , 1. and apply the previous argument to h in place of f 0 , defining f 1Ž x, c . as f 1Ž x, c . s c k r m ? f 2 Ž x m rc .. Then f 1 satisfies Ži. and discŽ f 1Ž x, 1.. s discŽ f 2 Ž x m .. is only divisible by primes which divide discŽ f 2 Ž x .. or m Ž0 is only a root when m s 1.. However, m divides k and so is relatively

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421

prime to k n y 1. Thus in all cases we are able to find a polynomial f 1Ž x, c . congruent to f 0 Ž x, c . mod p satisfying the requirements. Note that condition Živ. follows from the fact that the dnŽ1, c . polynomial for f 1 reduces to the dnŽ1, c . polynomial for f 0 Ž x, c . Žmod p .. Hence Fn, f 0Ž x, c . is absolutely irreducible over Fp , by Theorem 15 of w9x. This implies that Fn, f Ž x, c . is absolutely irreducible over k . For if Fn, f Ž x, c . factored over some finite extension L of k , then Fn, f Ž x, c . ' Fn, f 0Ž x, c . Žmod `. would factor in the residue class field of L Žmod P ., where P is some prime ideal of L dividing `, and so Fn, f 0Ž x, c . would factor in some finite extension of Fp . This argument also shows that dnŽ x . is irreducible over k s Q, by the proof of Corollary 1 in w9, Sect. 2x. Furthermore, condition Živ. implies conditions Žii. and Žiii. of Theorem 8 Žnote that deg D n, 1Ž c . s Ž k y 1. ? k ? f Ž n.rm is positive; see the proof of Theorem 11 in w9x.. Hence the Galois group of FnŽ x . is as given. Since dnŽ x . is irreducible over QŽ c ., its root v i generates the field Fi , so that GalŽ dnŽ x .rQŽ c .. s GalŽ LrQŽ c .. ( Sr , by Theorem 8. Case 2. Suppose k is a finite field. Let L be an algebraic number field and ` a prime ideal of L for which the residue class field of L mod ` is k . Let f 1Ž x . be a polynomial in Lw x x satisfying Ži. and Žii. and f 1Ž x . ' f Ž x . Žmod `.. Then conditions Žiii. and Živ. are satisfied by f 1 over L, and Case 1 shows that Fn, f 1Ž x . is absolutely irreducible over L. Now the proof of Theorem 15 in w9x can be applied, mutatis mutandis, to show that Fn, f Ž x . must be irreducible over k . ŽThe only use made of hypothesis ŽH. in that proof is the assertion that Fn, f 1Ž x . is absolutely irreducible.. The irreducibility of dnŽ x . over k follows again by the argument in w9, Sect. 2, proof of Corollary 1x. Conditions Žii. and Žiii. of Theorem 8 are implied by hypothesis Živ., as in Case 1. Finally, Proposition 10 of w9x shows that the ramification indices of the prime divisors of p` in the field K i are all m Žassuming Ži.., so this ramification is tame Žbecause of hypothesis Žiii.. and condition Živ. of Theorem 8 is also satisfied. This implies the assertions of Theorem B in this case. Case 3. Now let k be an arbitrary field and let k X be its prime field. Let Y be the set of coefficients of f Ž x, 1. and set DŽ Y . s discŽ f Ž x, 1.. ? discŽ dnŽ1, c ... By w22, Proposition 12, p. 35x, there is a finite specialization Y X of Y which is algebraic over k X and for which DŽ Y X . / 0. Then f Ž x, 1. specializes to a polynomial g Ž x, 1. which satisfies all the hypotheses of Theorem B, and whose coefficients belong either to an algebraic number field or to a finite field. The assertions of the theorem now follow from Cases 1 and 2 and the injective maps Gal Ž Fn Ž x, Y X . rk X Ž c . . ª Gal Ž Fn Ž x, Y . rk Ž c . . ª ZrnZ wr Sr

422

PATRICK MORTON

and Gal Ž Fn Ž x . rk Ž c . . ª Gal Ž Fn Ž x . rk Ž c . . ª ZrnZ wr Sr , where FnŽ x, Y X . is the specialization of FnŽ x . s FnŽ x, Y . under Y ª Y X . ŽSee w11x, Theorem 4.2 and Lemma 7.5x.. The same considerations apply to GalŽ dnŽ x .rk Ž c ... The following theorem is an immediate corollary of Theorem B. THEOREM 9. Ža. groups

Let f Ž x . s x k q c, with k G 2.

For any n G 1, FnŽ x . and dnŽ x . are irreducible o¨ er Q with Galois Gal Ž Fn Ž x . rQ Ž c . . ( ZrnZ wr Sr , Gal Ž dn Ž x . rQ Ž c . . ( Sr ,

where nr s deg FnŽ x .. Žb. If q s p s, where the prime p does not di¨ ide the integer k ? Ždisc dnŽ1, c .., then the abo¨ e isomorphisms are also ¨ alid o¨ er the finite field Fq : Gal Ž Fn Ž x . rFq Ž c . . ( ZrnZ wr Sr ,

Ž 7.

Gal Ž dn Ž x . rFq Ž c . . ( Sr .

Ž 8.

Proof. The map f Ž x . satisfies disc f Ž x, 1. s "k k , so conditions Ži. ] Žiii. of Theorem B are satisfied. Since the roots of dnŽ1, c . s 0 are exactly the ‘‘roots’’ of the hyperbolic components of the Mandelbrot set Žor its analogue, when k ) 2., condition Živ. is implied by the nontrivial and important result of Douady]Hubbard theory that these roots are distinct, for a given value of n Žsee w4x for k s 2, w17, Theorem 4.2.3x for k G 2 and w13, Proposition 3.2x.. This proves Ža.. Part Žb. follows immediately from Theorem B. An immediate consequence of Theorem 9Ža. Žor Theorem B. is that FnŽ x . is irreducible over CŽ c . with Galois group ZrnZ wr Sr . This result was first proven for k s 2 by Bousch w1x and for general k G 3 by Lau and Schleicher w7, 17x Žthough these authors use the semidirect product notation instead of the wreath product.. The result for dnŽ x . seems not to have been stated before. ŽSee w20x for the polynomials dnŽ x . when k s 2.. The approach we have taken can also be used to prove Theorem D stated in the Introduction. We restate the result here as Theorem 10 for ease of reference.

423

GALOIS GROUPS OF PERIODIC POINTS

THEOREM 10. Let f Ž x . s x k q c, with k G 2. For any n G 1 the Galois group of f n Ž x . y x o¨ er QŽ c . is the direct product Gal Ž f n Ž x . y xrQ Ž c . . s

m Gal Ž F Ž x . rQŽ c . . ( m ZrdZ d

d
d
wr S r d ,

Ž 9. where drd s deg Fd Ž x .. If PerŽ f . is the set of all periodic points of f in an algebraic closure of QŽ c ., then Gal Ž Q Ž Per Ž f . . rQ Ž c . . s

`

Ł Ž ZrdZ wr Sr . .

Ž 10 .

d

ds1

If MultŽ f . denotes the set of multipliers of all orbits of f in PerŽ f ., then Gal Ž Q Ž Mult Ž f . . rQ Ž c . . s

`

Ł Sr

ds1

Ž 11 .

d

and QŽPerŽ f .. is an abelian extension of QŽMultŽ f .. with Galois group Gal Ž Q Ž Per Ž f . . rQ Ž Mult Ž f . . . (

`

Ł Ž ZrdZ.

rd

.

Ž 12 .

ds1

Finally, QŽMultŽ f .. is unramified o¨ er the compositum Ł`ns 1QŽ D n , n Ž c . . at finite primes.

'

Proof. Let S d be the splitting field of Fd Ž x . over QŽ c . inside a given algebraic closure of QŽ c .. We will show that the S d are linearly disjoint over QŽ c ., i.e., that Sd l

Ł

X X d < n , d /d

S dX s Q Ž c . ,

for all divisors d of n Žthe product sign denotes the composite field of all the S dX .. From Ž1., Ž2. and the corollary to Proposition 3 we know that the ramified primes in S d , other than p` , are the zero divisors p b in QŽ c . of the linear polynomials c y b, where d d Ž1, b . s 0 and b is a root of a hyperbolic component corresponding to period d. The structure of the Mandelbrot set Žsee w3x and w17x. implies that the closures of two different hyperbolic components intersect in at most one point, which is then a root of the component of larger period. Hence components of different periods have different roots. Since the set of ramified primes of the composite field Ls

Ł

X X d < n , d /d

S dX

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PATRICK MORTON

is the union of the ramified primes of the individual fields S dX , it follows that L and S d have disjoint ramification, except at p` . Hence the field S d l L is unramified over QŽ c . except at p` , and is tamely ramified there. Lemma 7 gives therefore that S d l L s QŽ c .. The assertion Ž9. now follows from the Galois theory. To prove Ž10., note that the group GalŽQŽPerŽ f ..rQŽ c .. is the inverse limit of the groups GalŽ f n Ž x . y xrQŽ c .. ( md < n ZrdZ wr Sr d , as n ª `, where the maps defining the inverse limit are just projections; from this, Ž10. follows easily. The isomorphism Ž11. is proved in exactly the same way, by showing that the fields L d ; S d are linearly disjoint over QŽ c ., by noting that L d is generated over QŽ c . by multipliers, and then taking an inverse limit. The isomorphism Ž12. follows from Ž10., since any automorphism s of QŽPerŽ f .. which fixes QŽMultŽ f .. restricts to an automorphism of Nd ( ŽZrdZ. r d on the field S d Žcf. the remarks preceding Lemma 1.. Finally, Theorem A implies that the nth multiplier subfield L n is unramified over QŽ D n , n Ž c . . Žat finite primes.. Consequently, replacing the fields L d one at a time by the fields QŽ D d , d Ž c . . gives the following chain of unramified extensions:

'

'

Ln Q

Ł L drQ ž 'D n , n Ž c . / Ł L d ;

d-n

ž 'D

d-n

n, n

=

Ž c . / L ny1

Ł

Ld; . . . ;

ž 'D

d, d

d-ny1

Ł

1-dFn

Q

Ł

d-ny1

Ž c . / L1

L drQ

Ł

1FdFn

ž 'D

Q

n, n

ž 'D

Ž c . , 'D ny1, ny1 Ž c .

d, d

/

Ž c. / .

Putting this chain together shows that the extension Ł d F n L dr Ł d F nQŽ D d , d Ž c . . is unramified at finite primes. From this the last assertion of the theorem follows.

'

Exactly the same arguments may be used to prove Theorem C of the Introduction.

5. TWO EXAMPLES OVER Fp For the quadratic map f Ž x . s x 2 q c Theorem 9 shows that the isomorphisms Ž7. and Ž8. are true over Fp , for all but finitely many primes p. I conjecture in this case that Ž7. and Ž8. actually hold over any finite field of odd characteristic. This would imply the conjecture made in w10x that

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GALOIS GROUPS OF PERIODIC POINTS

Fn, f Ž x . is absolutely irreducible over Fp , for any odd prime p. Evidence for this conjecture is given in the following two examples. EXAMPLE. Consider f Ž x . s x 2 q c and n s 4. By w10, Lemma 1x, F4 Ž x . is absolutely irreducible over Fp for all odd primes p. In this case we have Žsee w13x.

d4 Ž 1, c . s D 4, 1 Ž c . D 4, 2 Ž c . D 4, 4 Ž c . s Ž 16 c 2 y 8c q 5 . Ž y4c y 5 . Ž 64 c 3 q 144c 2 q 108c q 135 . . The discriminant of d4Ž1, c . is y2 86 = 39 = 5 8 = 17 4 , and so Ž7. and Ž8. hold for n s 4 and all finite fields whose characteristic p / 2, 3, 5, 17. We note the factorizations D 4, 1 Ž c . D 4, 2 Ž c . D 4, 4 Ž c . ' Ž c 2 q c q 2 . ? Ž 2 c q 1 . ? c 3

Ž mod 3 . ,

D 4, 1 Ž c . D 4, 2 Ž c . D 4, 4 Ž c . ' c Ž c q 2 . ? c ? 4 c Ž c 2 q c q 2 .

Ž mod 5 . ,

D 4, 1 Ž c . D 4, 2 Ž c . D 4, 4 Ž c . s y Ž c q 2 . Ž c q 6 . ? 13 Ž c q 14 . ? 13 Ž c q 2 . Ž c 2 q 13c q 15 .

Ž mod 17 . .

In each case D 4, 1Ž c . satisfies condition Žii. of Theorem 8, so by the Corollary to Proposition 4 the Galois group of the splitting field S p of F4Ž x . over Fp Ž c . satisfies Gal Ž S prFp Ž c . . ( Ž Zr4Z.

wr Gal Ž L prFp Ž c . . , for p s 3, 5, 17.

On the other hand, we know from w10, Lemma 1x that the field L p is the splitting field over Fp Ž c . of the irreducible polynomial t4 Ž z . s z 3 q Ž4 c q 3. z q 4, whose roots are the traces of the three orbits of period 4. Since disc t4 Ž z . s y4D 4, 4 Ž c . is not a square Žmod p ., for any odd prime p, we see that GalŽ L prFp Ž c .. ( S3 and Gal Ž S prFp Ž c . . ( Ž Zr4Z.

wr S3 , for any prime p / 2.

ŽSee w10, Eq. Ž2.x or w20x for an explicit expression for d4 Ž x ... EXAMPLE. Consider f Ž x . s x 2 q c and n s 5. Here we have D 5, 1 Ž c . s 256c 4 q 64 c 3 q 16 c 2 y 36 c q 31, D 5, 5 Ž c . s 4,194,304c 11 q 32,505,856c 10 q 109,051,904c 9 q 223,084,544c 8 q 336,658,432 c7 q 402,464,768c 6 q 379,029,504c 5 q 299,949,056c 4 q 211,327,744c 3 q 120,117,312 c 2 q 62,799,428c q 28,629,151

426

PATRICK MORTON

Žsee w13, Table 1x.. A computation on Maple gives disc D 5, 1s2 24 = 5 7 = 112 , disc D 5, 5s2 274 = 312 = 3127 = 3701 = 4217 3 , Resultant Ž D 5, 1 , D 5, 5 . s2 116 = 113 = 319 = 86131. For all primes p not listed the Galois group of F5 Ž x, c . over Fp Ž c . is GalŽ S prFp Ž c .. ( Zr5Z wr S6 , by Theorem 9Žb.. We check that this also holds for all the odd primes which divide disc d 5 Ž1, c .. From w10x we know that F5 Ž x . is absolutely irreducible over Fp for all odd primes p Žsee also w9, Proposition 18x.. Let S s  3, 5, 11, 31, 3701, 4217, 861314 . First note for all primes p / 5 in S that D 5, 1Ž c . satisfies the hypotheses of Proposition 4. For example, the linear polynomials Ž c q 9., Ž c q 12., and Ž c q 4857. all divide D 5, 1Ž c . to exactly the first power modulo 11, 31, and 86131, respectively, but do not divide D 5, 5 Ž c . modulo those primes. Since these are the only primes p / 5 for which D 5, 1Ž c . could have a multiple factor or a factor in common with D 5, 5 Ž c ., this proves the assertion. Thus Gal Ž S prFp Ž c . . ( Zr5Z wr Gal Ž L prFp Ž c . . , for all p / 5 in S. Ž 13 . To show the same isomorphism for p s 5 requires more effort, since D 5, 1Ž c . ' Ž c q 1. 4 Žmod 5.. We show first that the zero divisor of Ž c q 1. ramifies in K irFi . To do this we consider the polynomial

t 5 Ž x, c . s x 6 q x 5 q Ž 3 q 11c . x 4 q Ž 11 q 18c . x 3 q Ž 44 q 19c q 19c 2 . x 2 q Ž 36 y 24 c q 17c 2 . x q Ž 32 q 28c q 40 c 2 q 9c 3 . , whose roots are the traces of the period 5 orbits, and whose discriminant is y65536Ž4 q 3c . 2D 5, 5 Ž c .. From w10x, t 5 Ž x, c . is absolutely irreducible over Fp , for any odd prime p, so an easy Galois theoretic argument shows that the roots of t 5 Ž x, c . generate the different fields Fi over Fp Ž c .. Now

t 5 Ž x, y1 . ' x Ž x 5 q x 4 q 2 x 3 q 3 x 2 q 4 x q 2 .

Ž mod 5, c q 1 . ,

where the quintic is irreducible Žmod 5., so that Ž c q 1. has a first degree prime divisor ` 1 and a fifth degree prime divisor ` 2 in Fi . Moreover,F4Ž x, y1. has Ž x q 2. as a fivefold multiple factor and five distinct irreducible quintics as factors Žmod 5.. ŽThe five quintics form a 5-cycle under the induced map of x ª x 2 y 1 over F5 , in the language of w2x; the cyclic is determined by one of the quintics, say x 5 q 4 x 3 q x 2 q 3..

GALOIS GROUPS OF PERIODIC POINTS

427

Since y2 is a fixed point of f Ž x . s x 2 q c s x 2 y 1 Žmod 5, c q 1.,  y2, y2, y2, y2, y24 is the orbit with trace 0, and it follows for any prime divisor P of ` 1 in K i , that P divides Ž x q 2.. However, Norm K Ž x q 2. s F5 Žy2, c ., which is exactly divisible by Ž c q 1. 2 Žmod 5.. From this fact it follows easily that ` 1 s P 5. Now the proof of Proposition 4 applies to the zero divisor of Ž c q 1., word for word, and shows that Ž13. also holds for p s 5. I claim now that the hypotheses of Proposition 5 are satisfied for all primes p in S. This can be checked by factoring D 5, 5 Ž c . Žmod p . on Maple and noting that in each case D 5, 5 Ž c . has a simple factor which does not divide D 5, 1Ž c .. ŽIt isn’t necessary to do this for p s 5.. Hence, GalŽ L prFp Ž c .. contains a transposition, for all p in S. Over F5 , D 5, 5 Ž c . is a product of an irreducible quartic and an irreducible seventh degree polynomial; the hypotheses of Proposition 6 are thus satisfied, conclusion Ži. of that proposition holds Žsince the ramification index at infinity is 2 and therefore tame; see the proof of Theorem 8., and GalŽ L 5rF5 Ž c .. s S6 . The same conclusion follows for the other primes in S by noting in each case that GalŽ L prFp Ž c .. contains a 6-cycle. This is because

t 5 Ž x, 0 . is irreducible mod 3, mod 11, mod 3701, and mod 86131, t 5 Ž x, 2 . is irreducible mod 31, and

t 5 Ž x, 3 . is irreducible mod 4217. This shows that GalŽ S prFp Ž c .. ( Zr5Z wr S6 , for all odd primes p. ACKNOWLEDGMENTS The results of this paper were influenced by a helpful conversation with R. W. K. Odoni and by an interesting talk given by Prof. Abhyankar, both of which occurred at the International Conference on Finite Fields in Glasgow, July, 1995. I am also grateful to Franco Vivaldi for several suggestions and clarifications, and to Mike Zieve and Siman Wang for their comments on algebraic number fields with no unramified extensions.

REFERENCES 1. T. Bousch, Sur quelques problemes de dynamique holomorphe, These, Universite de Paris-Sud, Centre d’Orsay, 1992. 2. A. Batra and P. Morton, Algebraic dynamics of polynomial maps on the algebraic closure of a finite field, I, Rocky Mountain J. Math. 24 Ž1994., 453]481.

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3. B. Branner, The Mandelbrot set, in ‘‘Chaos and Fractals’’ ŽR. L. Devaney and L. Keen, Eds.., pp. 75]105, AMS Publications, Providence, RI, 1989. 4. A. Douady and J. H. Hubbard, Etude dynamique des polynomes complexes, 1st and 2nd parts, Centre d’Orsay, Universite de Paris-Sud, 1984]1985. 5. M. Epkenhans, On double covers of the generalized symmetric group Z d wr S m as Galois groups over algebraic number fields K with m d subset K, J. Algebra 163 Ž1994., 404]423. 6. H. Hasse, ‘‘Zahlentheorie,’’ Akademie-Verlag, Berlin, 1969. 7. E. Lau and D. Schleicher, Internal addresses in the Mandelbrot set and irreducibility of polynomials, preprint 1994r19, Institute for Mathematical Sciences, SUNY Stony Brook, 1994. 8. P. Morton, Arithmetic properties of periodic points of quadratic maps, Acta Arith. LXII Ž1992., 343]372. 9. P. Morton, On certain algebraic curves related to polynomial maps, Comp. Math. 103 Ž1996., 319]350. 10. P. Morton, Arithmetic properties of periodic points of quadratic maps, II, preprint, Wellesley College, 1994. 11. P. Morton and P. Patel, The Galois theory of periodic points of polynomial maps, Proc. London Math. Soc. Ž 3 . 68 Ž1994., 225]263. 12. P. Morton and J. H. Silverman, Periodic points, multiplicities, and dynamical units, J. Reine Angew. Math. 461 Ž1995., 81]122. 13. P. Morton and F. Vivaldi, Bifurcations and discriminants for polynomials maps, Nonlinearity 8 Ž1995., 571]584. 14. J. Neukirch, ‘‘Algebraische Zahlentheorie,’’ Springer-Verlag, Berlin, 1992. 15. R. W. K. Odoni, The Galois theory of iterates and composites of polynomials, Proc. London Math. Soc. 51 Ž1985., 385]414. 16. R. W. K. Odoni, The Galois theory of iterated polynomials over arbitrary fields, preprint 94, University of Glasgow, 1994. 17. D. Schleicher, Internal addresses in the Mandelbrot set and irreducibility of polynomials, Ph.D. dissertation, Cornell University, 1994. 18. H. Stichtenoth, ‘‘Algebraic Function Fields and Codes,’’ Springer-Verlag Universitext, Berlin, 1993. 19. F. Vivaldi and S. Hatjispyros, Galois theory of periodic orbits of polynomial maps, Nonlinearity 5 Ž1992., 961]978. 20. F. Vivaldi and S. Hatjispyros, A family of rational zeta functions for the quadratic map, Nonlinearity 8 Ž1995., 321]332. 21. H. Wielandt, ‘‘Finite Permutation Groups,’’ Academic Press, New York, 1964. 22. A. Weil, ‘‘Foundations of Algebraic Geometry,’’ AMS Colloquium Publications, Vol. 29, Amer. Math. Soc., Providence, 1975.