Computing restrictions of ideals in finitely generated k -algebras by means of Buchberger’s algorithm

Journal of Symbolic Computation 41 (2006) 372–380 www.elsevier.com/locate/jsc

Computing restrictions of ideals in finitely generated k-algebras by means of Buchberger’s algorithm✩ Thomas Beth, J¨orn M¨uller-Quade, Rainer Steinwandt ∗ Institut f¨ur Algorithmen und Kognitive Systeme, Fakult¨at f¨ur Informatik, Am Fasanengarten 5, Universit¨at Karlsruhe, 76128 Karlsruhe, Germany Available online 2 November 2005

Abstract Gr¨obner bases can be used to solve various algorithmic problems in the context of finitely generated field extensions. One key idea is the computation of a certain kind of restriction of an ideal to a subring. With this restricted ideal many problems concerning function fields reduce to ideal theoretic problems which can be solved by means of Buchberger’s algorithm. In this contribution this approach is generalized to allow the computation of the restriction of an arbitrary ideal to a subring. c 2005 Elsevier Ltd. All rights reserved.  Keywords: Gr¨obner bases; Finitely generated function fields

1. Introduction Buchberger’s algorithm allows in particular for a constructive theory of ideals in polynomial rings, that led to a multitude of applications, one of which will be presented in this contribution. Many computational problems, especially concerning finitely generated field extensions, can be solved by the restriction of specific ideals to rings defined over a subfield. This paper presents a novel algorithm which allows one to restrict an arbitrary ideal I =  f1 , . . . , fs  in a residue ✩ Preliminary versions of this paper were presented in [Beth, T., M¨uller-Quade, J., Steinwandt, R., 2002. Computing restrictions of ideals in finitely generated k-algebras by means of Buchberger’s algorithm. In: Nakagawa, K. (Ed.), Proceedings of Symposium in Honor of Bruno Buchberger’s 60th Birthday; Logic, Mathematics and Computer Science: Interactions. LMCS 2002, pp. 39–47; Steinwandt, R., M¨uller-Quade, J., 2000. On restricting ideals in finitely generated k-algebras. In: Mulders, T. (Ed.), Proceedings of the Seventh Rhine Workshop on Computer Algebra. RWCA’00, pp. 119–124]. ∗ Corresponding address: Department of Mathematical Sciences, Florida Atlantic University, 777 Glades Road, Boca Raton 33431, FL, USA. Tel.: +1 561 297 3353; fax: +1 561 297 2436. E-mail address: [email protected] (R. Steinwandt).

c 2005 Elsevier Ltd. All rights reserved. 0747-7171/$ - see front matter  doi:10.1016/j.jsc.2005.09.010

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class ring k( x )[Z 1 , . . . , Z m ]/ q  to a subring k( g )[Z 1 , . . . , Z m ]/ q  defined over a subfield k( g) of k( x ) = k(x 1 , . . . , x n ), where q ∈ k( g )[Z 1 , . . . , Z m ]. So for the special case q = 0 we are dealing with a question on subalgebras of the polynomial ring k( x )[Z 1 , . . . , Z m ]. However, in contrast to Kapur and Madlener (1989), Robbiano and Sweedler (1990), for instance, here we do not focus on k( x )-subalgebras of the form k( x )[a1, . . . , ar ] with a1 , . . . , ar ∈ k( x )[Z 1 , . . . , Z m ]; instead we are interested in k-subalgebras k( g )[Z 1 , . . . , Z m ] that are obtained by restricting the field of coefficients k( x ). x )[Z 1 , . . . , Z m ]/ q , our Given finitely many generators f 1 , . . . , fs for an ideal I ⊆ k( q ; here we identify algorithm computes generators for the ideal I ∩ k( g )[Z 1 , . . . , Z m ]/ k( g)[Z 1 , . . . , Z m ]/ q  with its image under the natural embedding of k( g )[Z 1 , . . . , Z m ]/ q  into q . k( x )[Z 1 , . . . , Z m ]/ An instructive example is minimal polynomials: if one restricts the ideal Z − α ⊆ k(α)[Z ] to k[Z ] then Z −α∩ k[Z ] is a principal ideal and a monic generator of this ideal is the minimal polynomial of α over k. This work is motivated by the implications of Buchberger’s algorithm to finitely generated field extensions. Apart from the so-called tag variable approach (Kemper, 1993; Sweedler, 1993), many problems concerning field extensions can be solved by means of ideal restriction. One can employ a correspondence between the lattice of subfields k( g ) of a finitely generated field g )[Z 1 , . . . , Z n ]. k( x ) and a lattice of restricted ideals P(x )/ k(g) := Z 1 − x 1 , . . . , Z n − x n  ∩ k( This correspondence allows one to solve many problems concerning field extensions by means of constructive ideal theory and Buchberger’s algorithm (M¨uller-Quade and Steinwandt, 1999, 2000). Many characteristic properties of subfields directly translate to properties of the restricted ideals. For example, the transcendence degree of the extension k( g ) ≤ k( x ) equals the dimension n(x ) of the ideal P(x )/ k(g) , the polynomial n(Z 1 , . . . , Z n ) − d(x ) d(Z 1 , . . . , Z n ) reduces to zero

n(x ) g ), field extensions correspond modulo a Gr¨obner basis of P(x )/ k(g) iff d( x ) is contained in k( to ideal inclusions, and the coefficients of a reduced Gr¨obner basis of P(x )/ k(g) yield a canonical generating set of the field k( g ). For the specific ideals used in the above correspondence the restriction problem was solved in M¨uller-Quade and Steinwandt (1999, 2000). But the more general question of restricting an arbitrary ideal I to a finitely generated subfield was posed in M¨uller-Quade and Beth (1998a). In M¨uller-Quade and Beth (1998a) an algorithm to compute a generating set for the intersection of finitely generated extension fields has been sketched. Here we give a (counter-) example which shows that this algorithm does not work in general. In this approach to solve the field intersection problem the constructive restriction of more general ideals was introduced and used as a subroutine. The purpose of the present paper (see Steinwandt and M¨uller-Quade (2000) and Beth et al. (2002) for preliminary versions) is twofold. First we will solve the general ideal restriction problem and second we will show that the algorithm of M¨uller-Quade and Beth (1998a) cannot in all cases calculate the intersection of two finitely generated fields. Even though the general ideal restriction problem can be solved by Gr¨obner basis techniques, including computing the field of definition of an ideal, ideal saturation, primary decomposition, and ideal membership, the general field intersection problem remains an interesting open problem to solve.

2. Restricting ideals in finitely generated k-algebras To avoid ambiguities, we start by summarizing the notation that we use in what follows.

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 := K [Z 1 , . . . , Z m ] for the polynomial ring in the indeterminates • For K a field we write K [ Z] Z 1 , . . . , Z m over the field (of coefficients) K . • For K a field, I ⊆ K [ Z ] an ideal, and q ∈ K [ Z ] we denote by I : q ∞ the saturation of I w.r.t. q, i.e.,  : q μ · p ∈ I for some μ ∈ N}. I : q ∞ = { p ∈ K [ Z]  the set of terms in X , i.e., the set of all • For indeterminates X 1 , . . . , X u we denote by T ( X) u ν i products i=1 X i with ν1 , . . . , νu ∈ N0 . • k( x ) := k(x 1 , . . . , x n ) denotes a finitely generated (not necessarily algebraic) extension field of some ground field k. We assume computations in k( x ) to be effective and that k( x ) is represented as the quotient field of k[X 1 , . . . , X n ]/b1 , . . . , bt  where b is a finite system of generators of the ‘ideal of relations’  ∈ k[ X ] : a( x ) = 0}. P(x )/ k := {a( X) • k( g ) := k(g1 , . . . , gr ) denotes a subfield of k( x ) generated over the ground field k by x ). Note that the gi ’s can in general not be expressed as polynomials in x , g1 , . . . , gr ∈ k( and fractions are needed. • For an ideal I ⊆ k( x )[ Z ] we denote by kI the minimal field of definition of I. In other words, kI is the field generated over the prime field of k by the coefficients occurring in a reduced Gr¨obner basis of I (cf. M¨uller-Quade and R¨otteler, 1998; Robbiano and Sweedler, 1998).  denotes an arbitrary ideal whose minimal field of definition x )[ Z] • Q := q1 , . . . , qv  ⊆ k( kQ is contained in k( g ). As by computing a reduced Gr¨obner basis of Q a generating set  of Q can be derived, we assume w.l.o.g. q1 , . . . , qv ∈ kQ[ Z ] ⊆ k(  g )[ Z]. B ⊆ kQ[ Z]     • By πg : k( g )[ Z ] −→ k( g )[ Z]/ q  and πx : k( x )[ Z ] −→ k( x )[ Z]/ q  we denote the canonical residue class epimorphisms. • ι : k( g )[ Z ]/ q  −→ k( x )[ Z ]/ q  denotes the natural k( g )-algebra monomorphism that maps g )[ Z ]/ q  with the subring πg (Z i ) to πx (Z i ) (i = 1, . . . , m). In particular, we can identify k(   ι(k( g )[ Z]/ q ) of k( x )[ Z ]/ q . With this notation we can summarize the computational task to be solved in this section as follows.  q , Given a finite basis f1 , . . . , f s of an ideal I =  f 1 , . . . , f s  ⊆ k( x )[ Z]/  compute a finite generating set of the restricted ideal I ∩ ι(k( g )[ Z]/ q ).  of the ideal To compute this restriction, in a first step we determine a finite basis p ⊆ k( g )[ X]  ∈ k(  : a(  g )[ X] x ) = 0} = X 1 − x 1 , . . . , X n − x n  ∩ k( g )[ X]. P(x )/ k(g) := {a( X) This ideal will be a tool to compute the restriction of an ideal I; to this end we need the ideal P(x )/ k(g) in the X variables. In the introduction this ideal was formulated in the Z variables, as it was used as an example for a restriction.  reThe ideal P(x )/ k(g) will be important later, because modulo P(x )/ k(g) a polynomial c( X) duces to c( x ) iff c( x ) is contained in k( g ). We will later define an ideal H which contains all “denominator free” elements of I where the field elements x are replaced by variables X . This ideal H will also contain P(x )/ k(g) and we can, by computations within the ideal H, “replace”  by c( coefficients c( X) x ) without leaving the ideal iff c( x ) is contained in k( g ). Exactly the elements of H which correspond to elements outside of the restriction to be computed still contain coefficients in the variables X and can be removed from H by a simple variable elimination.

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The ideal P(x )/ k(g) can be computed by means of the following result (see M¨uller-Quade et al., 1998, Proposition 1).  and di ( x )/di ( x ) with n i , di ∈ k[ X] x ) = 0 Lemma 2.1. With the above notation let gi = n i ( (1 ≤ i ≤ r ). Then  ∞ r   − g1 ·dr ( X ), b1 , . . . , bt : di ( X ) . P(x )/ k(g) = n 1 ( X ) − g1 ·d1 ( X ), . . . , nr ( X) i=1

For effectively computing the saturation in Lemma 2.1 we can apply Becker and Weispfenning (1993, Proposition 6.37), for instance. Next, we fix for each generator f i of the ideal I a  i.e., for i = 1, . . . , s we have f i = πx ( f i ( representative fi ( X , Z ) ∈ k( X )[ Z], x , Z )). By  ∈ k[ X]  such that for i = 1, . . . , s “clearing denominators” we may select polynomials d˜i ( X) both d˜i ( x ) = 0 and  Z ) := d˜i ( X ) · f i ( X,  Z ) ∈ k[ X,  Z ] Fi = Fi ( X, hold. Now the essential tool we will use both for characterizing and for computing the restricted ideal I ∩ ι(k( g )[ Z ]/ q ) is the ideal     p, q  : h ∞ ⊆ k(  F, g )[ X , Z ]. H := h∈k(g)[ X ]\P(x )/k(g)

The ideal H is tailored such that for all “denominator free” elements f ( x , Z ) of  f the   polynomial f ( X, Z ) is contained in H. Furthermore, due to P(x )/ k(g) being contained in H a  for which c( coefficient c( X) x ) is contained in k( g ) can be “replaced by” c( x ) without leaving the ideal. Hence, all elements of I ∩ ι(k( g)[ Z ]/ q ) are contained in H and all other elements of H contain coefficients in X . By eliminating the variables X we obtain I ∩ ι(k( g )[ Z ]/ q ) from H, see Lemma 2.2.  Z ] The ideal H can in fact be written as a simple saturation. Exploiting the fact that k( g )[ X, is noetherian we obtain the following.  \ P(x )/ k(g) such that g )[ X] Remark. With the above notation, there is a polynomial h 0 ∈ k( ∞  H =  F, p, q  : h 0 .  Z ] is noetherian, there is a finite subset P ⊆ k(  with H = Proof. As k( g )[ X, g )[ X]  ∞  p, q  : h . For a finite sum one easily checks the inclusion  F, h∈P

   p, q  : h ∞ ⊆  F,  p, q  :  F, h∈P





∞ h

.

(1)

h∈P

  we have h 0 ∈ P(x )/ k(g) , because of the latter being a Thus, setting h 0 := h∈P h ∈ k( g )[ X],  p, q  : h 0 ∞ . Equality follows prime ideal. Moreover, from Eq. (1), we also know that H ⊆  F,  \ P(x )/ k(g) , i.e., h 0 is one of the summands occurring in the defining sum of g)[ X] from h 0 ∈ k( H.  By means of the ideal H, the restricted ideal I ∩ ι(k( g )[ Z ]/ q ) can now be characterized as follows.

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Lemma 2.2. With the above notation we have  q ) = (ι ◦ πg )(H ∩ k( I ∩ ι(k( g )[ Z]/ g)[ Z ]). g )[ X ] \ Proof. ‘⊇’: From the above remark we know that there exists a polynomial h 0 ∈ k(  p, q  : h 0 ∞ . P(x )/ k(g) with H =  F,  and a( Z)  ∈ (ι ◦ πg )−1 (a), i.e., for a suitable μ ∈ N we Now let a ∈ (ι ◦ πg )(H ∩ k( g )[ Z]) μ  ∈  F,  p, q  ⊆ k( have h 0 · a( Z) g )[ X , Z ]. Then, as h 0 ∈ P(x )/ k(g) , by specializing X i → x i    we obtain a( Z) ∈  f ( x , Z ), q  ⊆ k( x )[ Z ] resp.  ∈  f ⊆ k( πx (a( Z)) x )[ Z ]/ q .  ∈ k(  = πx (a( Z))  ∈  f = I. By assumption From a( Z) g)[ Z ] we conclude a = (ι ◦ πg )(a( Z))   a is contained in (ι ◦ πg )(k( g )[ Z ]), so we have a ∈ I ∩ ι(k( g )[ Z ]/ q ) as required.  Z ) ∈ k( X )[ Z]  of b = ‘⊆’: Let b ∈ I ∩ ι(k( g )[ Z ]/ q ), and fix a representation b( X,  Z ) − b( x , Z )). In particular, b( X, x , Z ) is contained in the kernel of the specialization (ι ◦ πg )(b( X i → x i , and because of T ( Z ) \ {1} being linearly independent over k( x ), there is a polynomial  ∈ k[ X]  with s( s( X) x ) = 0 and  Z ) − b(  Z ] ⊆ H. s( X ) · (b( X, x , Z )) ∈  p · k( g)[ X,

(2)

 q  such that b = As b is contained in the ideal I, there are ai ∈ k( x )[ Z]/ to representatives we can conclude  x , Z ) · fi ( x , Z ) + p0 b( x , Z ) = ai (



ai f i , and by passing

v  and ai ( for some p0 ∈  q  · k( x )[ Z] x , Z ) ∈ k( x )[ Z ]. Writing p0 = x , Z )q j with j =1 c j (  Z ) ∈ k( X)[  Z ] and choosing t ( X ) ∈ k[ X]  \ P(x )/ k appropriately we obtain c j ( X,      Z )Fi +  · b( X,  Z ) − c j ( X , Z )q j ∈ P(x )/ k(g) · k( g )[ X , Z ]. t ( X) ai ( X,

 ⊆k(g)[ X ]

g )[ X , Z ] being prime and t ( x ) = 0 we may conclude that From P(x )/ k(g) · k(     Z ) −  Z ) · Fi +  Z ) · q j ∈ P(x )/ k(g) · k(  Z ] ⊆ H. b( X, c j ( X, g )[ X, ai ( X,  q being contained in H, now also b( X,  Z ) ∈ H must hold. Because of F,   From (2) we therefore obtain s( X ) · b( x , Z ) ∈ H, and as H is saturated w.r.t. all polynomials  \ P(x )/ k(g) , we have b( x , Z ) ∈ H. Since b( x , Z ) ∈ k( g )[ Z ] this yields in k( g )[ X]  b = (ι ◦ πg )(b( x , Z )) ∈ (ι ◦ πg )(H ∩ k( g )[ Z]) as required.   q ) in Lemma 2.2 is From a computational point of view the characterization of I ∩ ι(k( g )[ Z]/ not really satisfying, as it does not give a hint on how to determine a basis of the ideal H. For computing such a finite set of generators for H, we can make use of the following remark.

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 p, q  = Remark. We keep the notation from Lemma 2.2. Let  F, primary decomposition. Then  H= Qi .

w

i=1

377

Qi be an irredundant

1≤i≤w Qi ∩k(g)[ X ]⊆P(x )/k(g)

Proof. According to Eisenbud (1995, Exercise 2.3) we have    Z ] ∩  F,  p, q  · k(  Z ] H = k( g)[ X, g )[ X,  P(x )/k(g) k(g)[ X]\  Z ] at  Z ] g )[ X, where as usual k( g )[ X,  P(x )/k(g) denotes the localization of k( k(g)[ X]\  \ P(x )/ k(g) . So the claim follows from Zariski and Samuel (1979, Chapter IV, k( g )[ X] Theorem 17).   ⊆ P(x )/ k(g) in the previous remark can be verified effectively The condition Qi ∩ k( g )[ X] by means of standard Gr¨obner basis techniques (cf., e.g., Buchberger, 1965, 1985; Trinks, 1978 and Becker and Weispfenning, 1993, Propositions 5.38 and 6.15). Moreover, if the  Z ] can be performed effectively then an irredundant primary required computations in k( g )[ X,  p, q  can be computed by means of the techniques described in Decker decomposition of  F, et al. (1999), Gianni et al. (1988), Seidenberg (1974) for instance. Finally, computing the  in Lemma 2.2 is a standard application of Gr¨obner basis techniques elimination ideal H∩k( g )[ Z] again, and as applying ι ◦ πg , which is essentially a change of the interpretation of the data computed, does not provide any difficulties, in summary we have the following. Theorem 2.1. We keep the above notation. Moreover, assume that an irredundant primary  p, q  ⊆ k(  Z ] can be computed effectively. Then a finite generating decomposition of  F, g )[ X, set of I ∩ ι(k( g )[ Z ]/ q ) can be computed effectively. We want to illustrate the method just described through two simple examples. Example. Let x be transcendental over Q, and consider the principal ideal I := Z 3 + Z 2 − x 3 − x 2  ⊆ Q(x)[Z ]. We want to compute the restriction I ∩ Q(x 2 )[Z ]. For this we first have to determine a basis of the ideal P(x)/Q(x 2) : obviously the minimal polynomial p := Z 2 − x 2 of x over Q(x 2 ) can be used here. Denoting the given generator of I by f1 , the corresponding polynomial F1 computes to Z 3 + Z 2 − X 3 − X 2 . As we are dealing with a polynomial ring we have Q = 0, and so in  p, q  is generated by our example the ideal  F, {Z 3 + Z 2 − X 3 − X 2 , X 2 − x 2 }. For example, by means of a computer algebra system like MAGMA (see Bosma et al., 1997) one  p, q  ⊆ Q(x 2)[Z ]: can determine the following irredundant primary decomposition of  F,  p, q  = Z − X, X 2 − x 2  ∩ Z 2 + Z X + Z + X + x 2 , X 2 − x 2   F,







=:Q1

=:Q2

By looking at the corresponding lexicographical Gr¨obner basis with Z > X we see that Qi ∩Q(x 2)[X] ⊆ P(x)/Q(x 2) holds for i = 1, 2. So in our example we have H = Q1 ∩Q2 , namely

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H = F1 , p. Computing the elimination ideal H ∩ Q(x 2 )[Z ] with a lexicographic Gr¨obner basis provides no further difficulties and yields I ∩ Q(x 2 )[Z ] = Z 6 + 2 · Z 5 + Z 4 − 2x 2 · Z 3 − 2x 2 · Z 2 − x 6 + x 4 . Example. Let x 1 , x 2 be algebraically independent over Q, and consider the ideal J := x 1 · Z 1 − Z 2 , x 2 · Z 2 − Z 3 , Z 32  ⊆ Q(x 1 , x 2 )[Z 1 , Z 2 , Z 3 ]. We want to compute the restriction J ∩ Q[Z 1 , Z 2 , Z 3 ]. The ideal P(x1,x2 )/Q is the zero ideal, and the polynomials F compute to F1 = X 1 · Z 1 − Z 2 , F2 = X 2 · Z 2 − Z 3 , F3 = Z 32 .  p, q  = Dealing again with a polynomial ring, we have Q = 0, and thus we obtain  F, F1 , F2 , F3  (⊆ Q[X 1 , X 2 , Z 1 , Z 2 , Z 3 ]). For example, by means of MAGMA one can determine  p, q  ⊆ the following irredundant primary decomposition I = Q1 ∩ Q2 ∩ Q3 of  F, Q[X 1 , X 2 , Z 1 , Z 2 , Z 3 ]: Q1 = X 12 , X 1 · Z 1 − Z 2 , X 1 · Z 2 , X 1 · Z 3 , X 2 · Z 2 − Z 3 , Z 22 , Z 2 · Z 3 , Z 32  Q2 = X 1 · Z 1 − Z 2 , X 22 , X 2 · Z 2 − Z 3 , X 2 · Z 3 , Z 32  Q3 = X 1 · Z 1 − Z 2 , X 2 · Z 2 − Z 3 , Z 12 , Z 1 · Z 2 , Z 1 · Z 3 , Z 22 , Z 2 · Z 3 , Z 32 . Only for i = 3 we have Qi ∩ Q[X 1 , X 2 ] = 0 (= P(x1 ,x2 )/Q ), and thus we obtain H = Q3 . By intersecting H with Q[Z 1, Z 2 , Z 3 ] we get (via Lemma 2.2) J ∩ Q[Z 1 , Z 2 , Z 3 ] = Z 12 , Z 1 · Z 2 , Z 1 · Z 3 , Z 22 , Z 2 · Z 3 , Z 32 . 3. A (counter-)example: Intersecting fields As described in M¨uller-Quade and Beth (1998a), an ideal restriction can be used to compute  of two subfields k(  ⊆ k( generators of the intersection k( g ) ∩ k(h) g ), k(h) x ): it is sufficient to find a basis of the ideal  X]  ⊆ (k(  X ]. g ) ∩ k(h))[ P(x )/ k(g) ∩k(h)[



(3)

 ⊆k(g)[ X]

Unfortunately, the method discussed in the previous section does not allow the computation of  is not a subfield of k( the intersection (3), as in general k(h) g ). In M¨uller-Quade and Beth (1998a) an algorithm for accomplishing this task was proposed, but a more detailed analysis shows that  x )[ X ] ∩ k(h)[X] which in general does not coincide it actually computes the ideal P(x )/ k(g) · k( with the ideal (3).  := Q(x 2 ) of k( x ) := Q(x). Example. Consider the two subfields k( g ) := Q(x 3 + x 2 ) and k(h) Then we know from the first example in the previous section that  ∩ k(h)[X]  x )[ X] = X 6 + 2 · X 5 + X 4 − 2x 2 · X 3 − 2x 2 · X 2 − x 6 + x 4 . P(x )/ k(g) · k(

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As adjoining the coefficients of a reduced Gr¨obner basis of this ideal to Q yields the field Q(x 2), the algorithm from M¨uller-Quade and Beth (1998a) yields Q(x 3 + x 2 ) ∩ Q(x 2 ) = Q(x 2 ), which is clearly wrong. So it remains an interesting open question whether the techniques described here can be extended in such a way that they allow the computation of a system of generators of the intersection of arbitrary finitely generated extension fields. References Becker, T., Weispfenning, V., 1993. Gr¨obner Bases: A Computational Approach to Commutative Algebra. In: Graduate Texts in Mathematics, vol. 141. Springer, New York (In cooperation with Heinz Kredel). Beth, T., M¨uller-Quade, J., Steinwandt, R., 2002. Computing restrictions of ideals in finitely generated k-algebras by means of Buchberger’s algorithm. In: Nakagawa, K. (Ed.) Proceedings of Symposium in Honor of Bruno Buchberger’s 60th Birthday; Logic, Mathematics and Computer Science: Interactions. LMCS 2002, pp. 39–47. Bosma, W., Cannon, J., Playoust, C., 1997. The Magma algebra system I: The user language. Journal of Symbolic Computation 24, 235–265. Buchberger, B., 1965. Ein Algorithmus zum Auffinden der Basiselemente des Restklassenringes nach einem nulldimensionalen Polynomideal (An algorithm for finding a basis for the residue class ring of a zero-dimensional polynomial ideal). Dissertation, Math. Inst., Universit¨at Innsbruck, Austria. Buchberger, B., 1985., Gr¨obner bases: An algorithmic method in polynomial ideal theory. In: Multidimensional Systems Theory. D. Reidel, Dordrecht, pp. 184–232. Decker, W., Greuel, G.-M., Pfister, G., 1999. Primary decomposition: Algorithms and comparisons. In: Matzat, B.H., Greuel, G.-M., Hiss, G. (Eds.), Algorithmic Algebra and Number Theory. Springer, pp. 187–220. Eisenbud, D., 1995. Commutative Algebra with a View Toward Algebraic Geometry. In: Graduate Texts in Mathematics, vol. 150. Springer, New York. Gianni, P., Trager, B., Zacharias, G., 1988. Gr¨obner bases and primary decomposition of polynomial ideals. Journal of Symbolic Computation 6, 149–167. Kapur, D., Madlener, K., 1989. A completion procedure for computing a canonical basis for a k-subalgebra. In: Computers and Mathematics. Springer, pp. 1–11. Kemper, G., 1993. An algorithm to determine properties of field extensions lying over a ground field. IWR Preprint 93-58, Heidelberg. M¨uller-Quade, J., Beth, T., 1998a. Computing the intersection of finitely generated fields. Poster presented at the International Symposium on Symbolic and Algebraic Computation, ISSAC’98. Abstract available in M¨uller-Quade and Beth (1998b). M¨uller-Quade, J., Beth, T., 1998b. Computing the intersection of finitely generated fields. SIGSAM Bulletin 32 (2), 62–62 (Abstract). M¨uller-Quade, J., R¨otteler, M., 1998. Deciding linear disjointness of finitely generated fields. In: Gloor, O. (Ed.), Proceedings of the 1998 International Symposium on Symbolic and Algebraic Computation. The Association for Computing Machinery, Inc. (ACM), pp. 153–160. M¨uller-Quade, J., Steinwandt, R., 1999. Basic algorithms for rational function fields. Journal of Symbolic Computation 27 (2), 143–170. M¨uller-Quade, J., Steinwandt, R., 2000. Gr¨obner bases applied to finitely generated field extensions. Journal of Symbolic Computation 30 (4), 469–490. M¨uller-Quade, J., Steinwandt, R., Beth, T., 1998. An application of Gr¨obner bases to the decomposition of rational mappings. In: Buchberger, B., Winkler, F. (Eds.), Gr¨obner Bases and Applications. In: Lecture Note Series, vol. 251. London Mathematical Society, Cambridge University Press, pp. 448–462. Robbiano, L., Sweedler, M., 1990. Subalgebra bases. In: Bruns, W., Simis, A. (Eds.), Commutative Algebra. In: Lecture Notes in Mathematics, vol. 1430. Springer, pp. 61–87. Robbiano, L., Sweedler, M., 1998. Ideal and subalgebra coefficients. Proceedings of the American Mathematical Society 126 (8), 2213–2219. Seidenberg, A., 1974. Constructions in algebra. Transactions of the American Mathematical Society 197, 273–313. Steinwandt, R., M¨uller-Quade, J., 2000. On restricting ideals in finitely generated k-algebras. In: Mulders, T. (Ed.) Proceedings of the Seventh Rhine Workshop on Computer Algebra. RWCA’00, pp. 119–124.

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