- Email: [email protected]
New examples of fractional dimensional semifield planes
Available online at www.sciencedirect.com
Electronic Notes in Discrete Mathematics 40 (2013) 71–75 www.elsevier.com/locate/endm
New examples of fractional dimensional semifield planes Minerva Cordero 1 Department of Mathematics University of Texas at Arlington Arlington, USA
LinLin Chen 2 Department of Mathematics Tuskegee University Tuskegee, USA
Abstract This is a short round up of the fractional dimension of finite semifields and finite semifield planes. We show that the classical Knuth binary semifields can never be fractional dimensional and a special class of the generalized Knuth binary semifields is fractional dimensional, even though these two classes of semifields are isotopic. Finally we show that the Albert semifield An (S), where n is any odd integer, does not contain GF (23 ) when (n, 3) = 1. Keywords: Finite semifields, semifield planes, fractional dimension
1 2
Email: [email protected] Email: [email protected]
1571-0653/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.endm.2013.05.014
72
1
M. Cordero, L. Chen / Electronic Notes in Discrete Mathematics 40 (2013) 71–75
Introduction
Let F be the finite field of order q n , where q is a prime power. It is well known that any subfield of F has order q m , where m is a divisor of n; the dimension n, of F = GF (q n ) over the subfield K = GF (q m ) is specified by logqm q n = m which is always an integer. One may more generally define the dimension of an arbitrary finite ternary ring with respect to any sub-ternary ring. Definition 1.1 Let D be a ternary ring of order n with a sub-ternary ring E of order m. Then the dimension of D relative to E is specified by dimE D := logm n; D is transcendental, fractional, or integer dimensional, relative to E, according to whether dimE D is transcendental, rational (but not an integer), or an integer. Similarly, if π is an affine plane of order n, with an affine subplane π0 of order m, the dimension of π relative to π0 is specified by logm n. In particular, π has transcendental, fractional, or integer dimension, relative to π0 , according to whether logm n is transcendental, rational (but not an integer), or an integer. Q In the 1950s, H. Neumann showed Q that any projective Hall plane of odd order contains Fano subplanes; hence is transcendental dimensional. In the late 1980s, de Resmini and her coworkers (Puccio, Leone) discovered other spectacular examples of transcendental dimensional planes. In particular, they showed that the Hughes plane of order 25 and also its derivative, the OstromRosati plane, admit subplanes of order 2 and 3. Further examples of transcendental affine planes have been obtained in this century by generalizing Neumann’s construction. For instance, Cordero-Jha proved the following: Theorem 1.2 To each Q square integer p2n , p an odd prime, there correspondsQan affine Hall plane that contains an affine Fano subplane Φ. Hence dimΦ = θ, a transcendental number. In this paper we are concern with the dimension of semifields and semifield planes. Suppose π is a translation plane of order pn , p prime, and π0 is an affine n subplane of π; then π0 has order pm for some integer m. Hence dimπ0 π = , m which is either an integer or a fractional number (not an integer). Until a few years ago only one example of a fractional dimensional plane was known: the Knuth semifield plane of order 32. In the last few years, Wene discovered other sporadic examples of fractional dimensional semifield planes again of characteristic 2. Recently Jha and Johnson, (see [4]), posed the following two problems about the fraction dimension of semifield planes. Problem 1.3 (a) There exist semifield planes of order 2r , for any odd
M. Cordero, L. Chen / Electronic Notes in Discrete Mathematics 40 (2013) 71–75
73
integer r that admit Desarguesian subplanes of order 22 . (b) There exist semifield planes of order 2t , for any integers t relatively prime to 3 that admit semifield subplanes of order 23 . The first problem was completely solved by Jha [2012], which implies that there exist infinitely many fractional dimensional semifield planes. However, for the second problem, no examples to support it have been found.
2
New Results
We focus on the dimension of three types of finite semifields of characteristic 2 in this paper: the classical Knuth binary semifields, the generalized Knuth binary semifields, and the Albert semifields. 2.1 The classical Knuth Binary Semifields are Non-Fractional Dimensional Let F = GF (2n ), n odd. For any x, y ∈ F , define a multiplication ◦ in F by x ◦ y = xy + (xT y + y T x)2 where T : F → GF (2) is the trace function. Then (F, +, ◦) is a pre-semifield. Choose 0 6= e ∈ F and define another new multiplication ∗ by x ∗ y = (x′ ◦ e) ∗ (e ◦ y ′) = x′ ◦ y ′ Then F = (F, +, ∗) is a semifield with identity e ◦ e. Obviously F is a commutative semifield. The following results concern sub-semifields of F and the nuclei of F . Theorem 2.1 Let F = (GF (2n ), +, ∗) be the classical Knuth binary semifield, n odd. Then for any integer k, 1 < k < n, there does not exist a sub-semifield of order 2k in F . Corollary 2.2 For the classical Knuth binary semifields, the left nucleus, the right nucleus, and middle nucleus are all GF (2); that is Nl = Nm = Nr = GF (2). It is uncommon for a semifield to admit no sub-semifields; we give a special name for such a semifield. Definition 2.3 Suppose S = (D, +, ∗) is a semifield. Then S is said to be simple if S has no proper sub-semifield. R´ ua proved that the classical Knuth binary semifield of order 2i , where i = 7, 9, 11, 13, is primitive. Then by our computation, we found that F is
74
M. Cordero, L. Chen / Electronic Notes in Discrete Mathematics 40 (2013) 71–75
also primitive when it has order 2i , where i = 15, 17, 19. Hence we make the following conjecture. Conjecture 2.4 Let S be a simple semifield, n-dimensional over its center GF (q). When q is large enough, S is both left and right primitive. 2.2 Dimension of the Generalized Knuth Binary Semifields Let F = GF (2n ), n ≥ 5 odd, and F0 = GF (2). Suppose GL(F, +) is the full group of F0 -linear bijections of the vector space F over F0 . Then for any x, y ∈ F define a multiplication ⊙ by x ⊙ y = xB y C + (xBT y C + y CT xB )2 where B, C ∈ GL(F, +). Then (F, +, ⊙) forms a pre-semifield. Choose any element e ∈ F ∗ , and define another new multiplication ∗ by (x ⊙ e) ∗ (e ⊙ y) = x ⊙ y,
∀ x, y ∈ F (e)
Then (F, +, ∗) is a semifield, denoted by FB,C (F ) = (F, +, ∗)e . The following results concern the dimension of the generalized Knuth binary semifields. Theorem 2.5 The generalized Knuth binary semifield of order 2n , for any odd integer n ∈ [5, 31], admits GF (22 ). Hence these semifields are fractional dimensional. (e) Theorem 2.6 For the generalized Knuth binary semifield FB,C (F ) = (F, +, ∗)e, if e ∈ Γ∗ = Ker(T ) − {0} and eB = eC , where B, C ∈ GL(F, +)Γ, then (a) (F, +, ∗)e is commutative and does not contain the finite field GF (22 ). (b) It does not contain GF (23 ) either when (n, 3) = 1. 2.3 Dimension of Albert Semifields This is a class of finite semifields with characteristic 2 due to Albert, who simplified the construction of Knuth binary semifields. Here is Albert’s idea: Let F = GF (2n ), n ≥ 3 odd, with multiplication juxtaposition and write F = S + ωGF (2), where S is an (n − 1)-dimensional vector space over GF (2) containing 1 and ω is an element not in S. Define s ∗ t = st, s ∗ ω = ω ∗ s = sω + s2 + s, ω ∗ ω = ω 2 + 1
M. Cordero, L. Chen / Electronic Notes in Discrete Mathematics 40 (2013) 71–75
75
for any s, t ∈ S. Then the vector space F with multiplication ∗ forms a proper semifield when n ≥ 5, denoted by An . Obviously Albert semifields are commutative. Wene proved that for any odd integer n, the Albert semifield An does not contain GF (22 ). Then we consider the existence of GF (23) in it and have the following result. Theorem 2.7 For any odd integer n with (n, 3) = 1, the Albert semifield An does not contain the finite field GF (23 ). Based on these results, we pose the following problem. Problem Let S be a commutative semifield. Is it possible for S to be fractional dimensional?
References [1] Cordero, M. and Jha, V., “Primitive semifields and fractional planes of order q 5 ”, Rendiconti Di Matematica, Serie VII, vol. 30, Roma, pp. 1-21, 2010 [2] Knuth, D. E.,“Finite semifields and projective planes”, J. Algebra, 2, pp. 182217, 1965. [3] Knuth, D. E.,“A class of projective planes”, Trans. Amer. Math. Soc., 115, pp. 541-549, 1965. [4] R´ ua, I.F., “Primitive and non primitive finite semifields”, Comm. Algebra, 22, pp. 791–803, 2004. [5] Jha, V. and Johnson, N. L., “The dimension of a subplane of a translation plane”, Bull. Belg. Math. Soc. Simon Stevin, 17, n. 3, pp. 523-535, 2010. [6] Leone, O. and de Resmini, M. J., “Subplanes of the derived Hughes planes of order 25”, Simon Stevin, 67, pp. 289-322, 1993. [7] Neumann, H., “On some finite non-desarguesian planes,” Arch. Math., 6, pp. 36-40, 1955. [8] Puccio, L. and de Resmini, M. J., “Subplanes of the Hughes planes of order 25”, Arch. Math. (Basel), pp. 151-165, 1987. [9] Wene, G. P., “On the multiplicative structure of finite division rings”, Aequationes Math., 41, pp. 463-477, 1991. [10] Wene, G. P., “Semifields: Two Constructions”, Presented to Students and Faculty of The University of Texas at Arlington, 2012.
Electronic Notes in Discrete Mathematics 40 (2013) 71–75 www.elsevier.com/locate/endm
New examples of fractional dimensional semifield planes Minerva Cordero 1 Department of Mathematics University of Texas at Arlington Arlington, USA
LinLin Chen 2 Department of Mathematics Tuskegee University Tuskegee, USA
Abstract This is a short round up of the fractional dimension of finite semifields and finite semifield planes. We show that the classical Knuth binary semifields can never be fractional dimensional and a special class of the generalized Knuth binary semifields is fractional dimensional, even though these two classes of semifields are isotopic. Finally we show that the Albert semifield An (S), where n is any odd integer, does not contain GF (23 ) when (n, 3) = 1. Keywords: Finite semifields, semifield planes, fractional dimension
1 2
Email: [email protected] Email: [email protected]
1571-0653/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.endm.2013.05.014
72
1
M. Cordero, L. Chen / Electronic Notes in Discrete Mathematics 40 (2013) 71–75
Introduction
Let F be the finite field of order q n , where q is a prime power. It is well known that any subfield of F has order q m , where m is a divisor of n; the dimension n, of F = GF (q n ) over the subfield K = GF (q m ) is specified by logqm q n = m which is always an integer. One may more generally define the dimension of an arbitrary finite ternary ring with respect to any sub-ternary ring. Definition 1.1 Let D be a ternary ring of order n with a sub-ternary ring E of order m. Then the dimension of D relative to E is specified by dimE D := logm n; D is transcendental, fractional, or integer dimensional, relative to E, according to whether dimE D is transcendental, rational (but not an integer), or an integer. Similarly, if π is an affine plane of order n, with an affine subplane π0 of order m, the dimension of π relative to π0 is specified by logm n. In particular, π has transcendental, fractional, or integer dimension, relative to π0 , according to whether logm n is transcendental, rational (but not an integer), or an integer. Q In the 1950s, H. Neumann showed Q that any projective Hall plane of odd order contains Fano subplanes; hence is transcendental dimensional. In the late 1980s, de Resmini and her coworkers (Puccio, Leone) discovered other spectacular examples of transcendental dimensional planes. In particular, they showed that the Hughes plane of order 25 and also its derivative, the OstromRosati plane, admit subplanes of order 2 and 3. Further examples of transcendental affine planes have been obtained in this century by generalizing Neumann’s construction. For instance, Cordero-Jha proved the following: Theorem 1.2 To each Q square integer p2n , p an odd prime, there correspondsQan affine Hall plane that contains an affine Fano subplane Φ. Hence dimΦ = θ, a transcendental number. In this paper we are concern with the dimension of semifields and semifield planes. Suppose π is a translation plane of order pn , p prime, and π0 is an affine n subplane of π; then π0 has order pm for some integer m. Hence dimπ0 π = , m which is either an integer or a fractional number (not an integer). Until a few years ago only one example of a fractional dimensional plane was known: the Knuth semifield plane of order 32. In the last few years, Wene discovered other sporadic examples of fractional dimensional semifield planes again of characteristic 2. Recently Jha and Johnson, (see [4]), posed the following two problems about the fraction dimension of semifield planes. Problem 1.3 (a) There exist semifield planes of order 2r , for any odd
M. Cordero, L. Chen / Electronic Notes in Discrete Mathematics 40 (2013) 71–75
73
integer r that admit Desarguesian subplanes of order 22 . (b) There exist semifield planes of order 2t , for any integers t relatively prime to 3 that admit semifield subplanes of order 23 . The first problem was completely solved by Jha [2012], which implies that there exist infinitely many fractional dimensional semifield planes. However, for the second problem, no examples to support it have been found.
2
New Results
We focus on the dimension of three types of finite semifields of characteristic 2 in this paper: the classical Knuth binary semifields, the generalized Knuth binary semifields, and the Albert semifields. 2.1 The classical Knuth Binary Semifields are Non-Fractional Dimensional Let F = GF (2n ), n odd. For any x, y ∈ F , define a multiplication ◦ in F by x ◦ y = xy + (xT y + y T x)2 where T : F → GF (2) is the trace function. Then (F, +, ◦) is a pre-semifield. Choose 0 6= e ∈ F and define another new multiplication ∗ by x ∗ y = (x′ ◦ e) ∗ (e ◦ y ′) = x′ ◦ y ′ Then F = (F, +, ∗) is a semifield with identity e ◦ e. Obviously F is a commutative semifield. The following results concern sub-semifields of F and the nuclei of F . Theorem 2.1 Let F = (GF (2n ), +, ∗) be the classical Knuth binary semifield, n odd. Then for any integer k, 1 < k < n, there does not exist a sub-semifield of order 2k in F . Corollary 2.2 For the classical Knuth binary semifields, the left nucleus, the right nucleus, and middle nucleus are all GF (2); that is Nl = Nm = Nr = GF (2). It is uncommon for a semifield to admit no sub-semifields; we give a special name for such a semifield. Definition 2.3 Suppose S = (D, +, ∗) is a semifield. Then S is said to be simple if S has no proper sub-semifield. R´ ua proved that the classical Knuth binary semifield of order 2i , where i = 7, 9, 11, 13, is primitive. Then by our computation, we found that F is
74
M. Cordero, L. Chen / Electronic Notes in Discrete Mathematics 40 (2013) 71–75
also primitive when it has order 2i , where i = 15, 17, 19. Hence we make the following conjecture. Conjecture 2.4 Let S be a simple semifield, n-dimensional over its center GF (q). When q is large enough, S is both left and right primitive. 2.2 Dimension of the Generalized Knuth Binary Semifields Let F = GF (2n ), n ≥ 5 odd, and F0 = GF (2). Suppose GL(F, +) is the full group of F0 -linear bijections of the vector space F over F0 . Then for any x, y ∈ F define a multiplication ⊙ by x ⊙ y = xB y C + (xBT y C + y CT xB )2 where B, C ∈ GL(F, +). Then (F, +, ⊙) forms a pre-semifield. Choose any element e ∈ F ∗ , and define another new multiplication ∗ by (x ⊙ e) ∗ (e ⊙ y) = x ⊙ y,
∀ x, y ∈ F (e)
Then (F, +, ∗) is a semifield, denoted by FB,C (F ) = (F, +, ∗)e . The following results concern the dimension of the generalized Knuth binary semifields. Theorem 2.5 The generalized Knuth binary semifield of order 2n , for any odd integer n ∈ [5, 31], admits GF (22 ). Hence these semifields are fractional dimensional. (e) Theorem 2.6 For the generalized Knuth binary semifield FB,C (F ) = (F, +, ∗)e, if e ∈ Γ∗ = Ker(T ) − {0} and eB = eC , where B, C ∈ GL(F, +)Γ, then (a) (F, +, ∗)e is commutative and does not contain the finite field GF (22 ). (b) It does not contain GF (23 ) either when (n, 3) = 1. 2.3 Dimension of Albert Semifields This is a class of finite semifields with characteristic 2 due to Albert, who simplified the construction of Knuth binary semifields. Here is Albert’s idea: Let F = GF (2n ), n ≥ 3 odd, with multiplication juxtaposition and write F = S + ωGF (2), where S is an (n − 1)-dimensional vector space over GF (2) containing 1 and ω is an element not in S. Define s ∗ t = st, s ∗ ω = ω ∗ s = sω + s2 + s, ω ∗ ω = ω 2 + 1
M. Cordero, L. Chen / Electronic Notes in Discrete Mathematics 40 (2013) 71–75
75
for any s, t ∈ S. Then the vector space F with multiplication ∗ forms a proper semifield when n ≥ 5, denoted by An . Obviously Albert semifields are commutative. Wene proved that for any odd integer n, the Albert semifield An does not contain GF (22 ). Then we consider the existence of GF (23) in it and have the following result. Theorem 2.7 For any odd integer n with (n, 3) = 1, the Albert semifield An does not contain the finite field GF (23 ). Based on these results, we pose the following problem. Problem Let S be a commutative semifield. Is it possible for S to be fractional dimensional?
References [1] Cordero, M. and Jha, V., “Primitive semifields and fractional planes of order q 5 ”, Rendiconti Di Matematica, Serie VII, vol. 30, Roma, pp. 1-21, 2010 [2] Knuth, D. E.,“Finite semifields and projective planes”, J. Algebra, 2, pp. 182217, 1965. [3] Knuth, D. E.,“A class of projective planes”, Trans. Amer. Math. Soc., 115, pp. 541-549, 1965. [4] R´ ua, I.F., “Primitive and non primitive finite semifields”, Comm. Algebra, 22, pp. 791–803, 2004. [5] Jha, V. and Johnson, N. L., “The dimension of a subplane of a translation plane”, Bull. Belg. Math. Soc. Simon Stevin, 17, n. 3, pp. 523-535, 2010. [6] Leone, O. and de Resmini, M. J., “Subplanes of the derived Hughes planes of order 25”, Simon Stevin, 67, pp. 289-322, 1993. [7] Neumann, H., “On some finite non-desarguesian planes,” Arch. Math., 6, pp. 36-40, 1955. [8] Puccio, L. and de Resmini, M. J., “Subplanes of the Hughes planes of order 25”, Arch. Math. (Basel), pp. 151-165, 1987. [9] Wene, G. P., “On the multiplicative structure of finite division rings”, Aequationes Math., 41, pp. 463-477, 1991. [10] Wene, G. P., “Semifields: Two Constructions”, Presented to Students and Faculty of The University of Texas at Arlington, 2012.