On finite semifelds of prime degree. equivalence classifcation of subspaces of invertible matrices

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1 On finite semifields of prime degree and the equivalence classifcation of subspaces of invertible matrices John Sheekey Rod Gow Claude Shannon Institute School of Mathematical Science University College Dublin 15 July 2009 / 9th International Conference on Finite Fields and their Applications

2 Outline 1 Semifields and presemifields Primitive elements 2 3

3 Outline Semifields and presemifields Primitive elements 1 Semifields and presemifields Primitive elements 2 3

4 Semifields and presemifields Primitive elements Definition A finite presemifield S = (V, +, ) is a (not necessarily associative) division algebra over a finite field F q. Definition A presemifield is called a semifield if it contains a multiplicative identity. Multiplication is F q bilinear. S has no zero divisors. F q is called the associative centre of S

5 Semifields and presemifields Primitive elements Definition Two presemifields (V, +, ) and (V, +, ) are said to be isotopic if there exist invertible linear transformations A, B, C : V V such that A(x y) = B(x) C(y) for all x, y V Every presemifield is isotopic to a semifield There are many constructions, but classification is difficult, and only known up to size 125

6 Semifields and presemifields Primitive elements Example (Albert s twisted fields) Define a new multiplication on F q n by x y := xy cx α y β where α, β are F q -automorphisms of F q n, and c a α 1 b β 1 for any a, b F q n. Then S := (F q n, +, ) is a presemifield. Menichetti (1996) showed that all semifields of prime dimension over their associative centre F q, for q large enough, are twisted fields.

7 Outline Semifields and presemifields Primitive elements 1 Semifields and presemifields Primitive elements 2 3

8 Semifields and presemifields Primitive elements Definition An element a of S is said to be left primitive if the elements a (i := aa (i 1 exhaust the non-zero elements of S. Most known semifields contain a primitive element. However, there exist semifields of size 32 and 64 containing no left or right primitive elements (Hentzel & Rua 2007). We aim to prove that all semifields of prime dimension over their associative centre F q, for q large enough, contain both left and right primitive elements.

9 Outline 1 Semifields and presemifields Primitive elements 2 3

10 For any field, the maximum dimension of a subspace of n n invertibles is n. Definition We say that two subspaces S and T are equivalent if there exist invertible matrices X and Y such that T = {XAY A S} We will see that this is related to the classification of presemifields up to isotopy.

11 Lemma The matrices of left-multiplication of the elements of a semifield S form a n-dimensional subspace of invertible matrices over F q. Proof. For each non-zero element a S define an element of M n (F q ) by L a (x) := ax x S Then L a is clearly invertible, for L a (x) = 0 ax = 0 x = 0. Then the set L S := {L a a S} is a n-dimensional subspace of n n invertibles.

12 Lemma Every basis for a n-dimensional subspace of n n invertible matrices defines a unique presemifield Proof. Let B = {E 1, E 2,..., E n } be such a basis. Define a presemifield S B := (F n, +, ) by Then S is a presemifield x y := (x 1 E 1 + x 2 E x n E n )y Note that if S is a semifield, we take E 1 = I.

13 Lemma Two presemifields are isotopic if and only if their associated subspaces are equivalent, i.e. S T L S L T

14 Outline 1 Semifields and presemifields Primitive elements 2 3

15 Lemma An element a is left primitive if and only if its matrix of left multiplication A has primitive characteristic polynomial. Proof. Note that a (i = A i e Suppose a is primitive. Then A i e e 0 < i < q n 1 Hence A i I, so A must have primitive characteristic polynomial.

16 Proof. Suppose now that A has primitive characteristic polynomial. Then F(A) = F q n Hence if 0 < i < q n 1, A i I is a non-zero element of F(A), and is hence invertible. Therefore, A i e e, and so a is a left primitive element. Hence it suffices to show that every subspace of n n invertibles contains a matrix with primitive characteristic polynomial.

17 Outline 1 Semifields and presemifields Primitive elements 2 3

18 Let {E 1 = I, E 2,..., E n } be a basis for a subspace of invertible matrices, obtained from a semifield S. Define a polynomial f S F q [x 1, x 2,..., x n ] by f S (x 1, x 2,..., x n ) := det(x 1 I + x 2 E x n E n ) Then f S is a degree n homogeneous polynomial in n variables, which has no non-trivial zeros in F n q.

19 Theorem (Chevalley-Warning) Let f be a homogeneous polynomial over F in m variables of degree d. Then if d < m, f has a non-trivial zero. Lemma The product of two polynomials g and h is homogeneous if and only if g and h are both homogeneous. Corollary Let f be a homogeneous polynomial over F in m variables of degree m with no non-trivial zeroes. Then f is irreducible over F.

20 Definition A polynomial in F[x 1,..., x m ] is said to be absolutely irreducible if it is irreducible in every finite algebraic extension of F (or, equivalently, is irreducible over F, the algebraic closure of F). Theorem (Lang-Weil) Let f be an absolutely irreducible homogeneous polynomial over F q in m variables of degree d. Then N, the number of zeros of f in F n q, satisfies N q m 1 (d 1)(d 2)q m Cq m 2 for some C which does not depend on q.

21 Cafure-Matera gave an explicit bound: Theorem (Cafure, Matera (2006)) Let f be as above. Then N q m 1 (d 1)(d 2)q m d 13 3 q m 2 Hence the polynomial arising from a subspace of invertibles is irreducible, but, for q large enough, cannot be absolutely irreducible.

22 We will show that, for n prime, this means that it must be a norm form : Definition A homogeneous polynomial f in n variables is said to be a norm form if f = gg σ... g σn 1 = n 1 i=0 g σi where g is a linear form in F q n[x 1,..., x n ], and σ is the Frobenuis automorphism.

23 Lemma Let f be an irreducible homogeneous polynomial over F q in n variables of degree n. Suppose f has a non-trivial factor g of degree r, irreducible over F q m, for some m. Then n = mr, and f = λgg σ... g σm 1 for some λ F q.

24 Proof. Let f = gh in F q m[x 1,..., x n ]. Then f σi = f = g σi h σi for all i. Since g is irreducible, g σi is irreducible for all i. Hence G := gg σ... g σm 1 divides f. (UFD) But G σ = G, and so G F q [x 1,..., x n ]. But f is irreducible in F q [x 1,..., x n ], therefore f = λg for some λ F q as claimed.

25 Outline 1 Semifields and presemifields Primitive elements 2 3

26 Hence we have the following: Lemma Let S be a presemifield of prime dimension n over it s associative centre F q. Then for q large enough, f S is a norm form over F q n Proof. We saw that f S is irreducible, but not absolutely irreducible. Hence f S has a divisor g of degree r over F q m, where mr = n and r n. But n is prime, and hence m = n and r = 1, i.e. f S is a norm form over F q n

27 Hence we can prove our main result... Theorem Let S be a semifield of prime degree n over it s associative centre F = F q. Then if q is large enough, S contains a primitive element. Proof. We have seen that f S must be a norm form in F q n, i.e. n 1 f S (x 1, x 2,..., x n ) = (x 1 + a 2 x a n x n ) σi i=0 for some a 2,..., a n F q n. Let α = a 1 λ 1 + a 2 λ a n λ n be a primitive element of F q n. Let A = λ 1 I λ n E n.

28 Proof. Then char(a) = det(xi A) = f S (x λ 1, λ 2,..., λ n ) n 1 = (x α) σi i=0 which is a primitive polynomial by definition. Example If n = 5 and q > 6296, then S contains a left- and right-primitive element. (for n = 3, every semifield contains primitive elements by a result of Rúa.)

29 We have shown that if q is large enough and n is prime, then any semifield of size q n with associative centre F q contains a right and left primitive element

30 References D.E. Knuth, Finite Semifields and Projective Planes, J. Algebra 2 (1965) Kantor, Finite Semifields, Finite Geometries, Groups, and Computation (2006) Rua, Primitive and non-primitive Finite Semifields, Communications in Algebra (2004) Menichetti, n-dimensional algebras over a field with a cyclic extension of degree n, Geom. Ded. (1996) Cafure Matera, Improved explicit estimates on the number of solutions of equations over a finite field, Finite Fields Appl. (2006)

COMMUTATIVE PRESEMIFIELDS AND SEMIFIELDS

COMMUTATIVE PRESEMIFIELDS AND SEMIFIELDS ROBERT S. COULTER AND MARIE HENDERSON Abstract. Strong conditions are derived for when two commutative presemifields are isotopic. It is then shown that any commutative