A NEW APPROACH to FINITE SEMIFIELDS 1. Introduction A

A NEW APPROACH TO FINITE SEMIFIELDS

SIMEON BALL, GARY EBERT, AND MICHEL LAVRAUW

Abstract. A finite semifield is shown to be equivalent to the existence of a particular geometric configuration of subspaces with respect to a Desarguesian spread in a finite dimensional vector space over a finite field. In 1965 Knuth [7] showed that each finite semifield generates in total six (not necessarily isotopic) semifields. In certain cases, the geometric interpretation obtained here allows us to construct another six semifields, hence, in total twelve (not necessarily isotopic) semifields are obtained from a given finite semifield.

1. Introduction

A semifield is an algebra satisfying the axioms for a skew field except possibly associativity of multiplication. The study of semifields started about a century ago (see [4]) and used to be called non-associative division rings or distributive quasifields. Following Knuth [7] and the recent literature we will use the name semifield. Throughout this paper the term semifield will always be used to denote a finite semifield. Semifields can be used to construct certain translation planes (called semifield planes) and two semifield planes are isomorphic if and only if the corresponding semifields are isotopic. Associated with a translation plane there is a spread by the so-called André-Bruck-Bose construction. The spread corresponding to a semifield plane is called a semifield spread. In the next section we give a geometric construction of a semifield spread (and hence a semifield) starting from a particular configuration of subspaces with respect to a Desarguesian spread. In Section 3 we show that every semifield can be constructed in this way. The construction of the twelve semifields from a given semifield is given in Section 4. For more on spreads, translation planes and isotopy we refer to [1], [3], and [5].

2. A geometric construction of finite semifields

If V is a vector space of rank d over a finite field with q elements, a spread of V is a set 0 0 S of subspaces W1,...,Wk of V , all of the same rank d , 1 ≤ d ≤ d, such that every non-zero vector of V is contained in exactly one of the elements of S. It follows that d0|d and that |S| = k = (qd − 1)/(qd0 − 1) (see [3]). In the case that d is even and d0 = d/2 we call a spread of V a semifield spread if there exists an element S of this spread and a group G of semilinear automorphisms of V such that G fixes S pointwise and acts transitively on the other elements of the spread.

Date: 3 October 2005. The first author acknowledges the support of the Ministerio de Ciencia y Tecnologia, España. During this research the third author was supported by a VENI grant, part of the Innovational Research Incentives Scheme of the Netherlands Organisation for Scientific Research (NWO). 1 2 SIMEON BALL, GARY EBERT, AND MICHEL LAVRAUW

n Let F be the finite field of q elements and let F0 be the subfield of q elements. Let V1 ⊕Vr, 2 ≤ r ≤ n be a vector space of rank r + 1 over F, where Vi is a subspace of rank i over F. Consider V1 ⊕ Vr as a vector space of rank (r + 1)n over F0 and let S(V1 ⊕ Vr) be the spread of subspaces of rank n over F0 arising from the spread of subspaces of rank 1 over F. Such a spread (i.e. arising from a spread of subspaces of rank 1 over some extension field) is called a Desarguesian spread. A Desarguesian spread has the property that it induces a Desarguesian spread in every subspace spanned by elements of the spread. Remark 2.1. The incidence structure constructed from a spread of t-spaces of a vector space of rank rt, generalising the André-Bruck-Boseconstruction of a translation plane, is a 2 − (qrt, qt, 1)-design with parallelism. This design is isomorphic to the incidence structure of points and lines of AG(r, qt) (the r-dimensional affine geometry over the finite field with qt elements) if and only if the spread is a Desarguesian spread. In the literature such spreads are sometimes called normal or geometric. The above motivates our choice to use the word Desarguesian.

For any subset T of V1 ⊕ Vr deﬁne

B(T ) = {S ∈ S(V1 ⊕ Vr) | T ∩ S 6= {0}}.

Let U and W be F0-subspaces of Vr of rank n and rank (r − 1)n, respectively, with the property that B(U) ∩ B(W ) = ∅.

Let 0 6= v ∈ V1, consider the quotient geometry (V1 ⊕ Vr)/W and deﬁne S(U, W ) = {hS,W i/W | S ∈ B(hv, Ui)}.

Theorem 2.2. S(U, W ) is a semiﬁeld spread of (V1 ⊕ Vr)/W .

Proof. First we show that S(U, W ) is a spread of (V1 ⊕ Vr)/W . Note that (V1 ⊕ Vr)/W is a vectorspace of rank 2n and the elements of S(U, W ) are subspaces of rank n. Suppose that hR,W i/W and hS,W i/W are elements of S(U, W ) \{Vr/W } with R and S distinct elements of B(hv, Ui) \ B(U) such that hR,W i/W and hS,W i/W have a non-trivial intersection. This implies that the subspace hR,S,W i has rank at most (r+1)n−1, hence hR,Si∩W 6= {0}. Now W ⊂ Vr and so hR,Si∩W ⊂ Vr and, since R and S are subspaces belonging to the Desarguesian spread S(V1 ⊕ Vr), hR,Si intersects Vr in an element of the spread. Moreover this element of the spread belongs to B(U) since R and S are elements of B(hv, Ui) \ B(U). Therefore there is an element of B(U) having a non-trivial intersection with W , contradicting B(U) ∩ B(W ) = ∅. This shows that any two elements of S(U, W ) \{Vr/W } have trivial intersection. Clearly also Vr/W has trivial intersection with every element of S(U, W ) \{Vr/W }, and hence any two elements of S(U, W ) have trivial intersection. In order to prove that S(U, W ) is a spread it now suﬃces to check that n S(U, W ) contains q + 1 elements. Clearly |{hS,W i/W | S ∈ B(U)}| = |{Vr/W }| = 1 and hence we have to show that {hS,W i/W | S ∈ B(hv, Ui) \ B(U)} has qn elements. This follows immediately since an element S of B(hv, Ui) \ B(U) which intersects hv, Ui in a subspace of rank ≥ 2 also intersects U and hence by deﬁnition belongs to B(U), a contradiction. Hence every element of B(hv, Ui) \ B(U) intersects hv, Ui in a subspace of rank 1. A simple count allows us to conclude that S(U, W ) is a spread of (V1 ⊕ Vr)/W . A NEW APPROACH TO FINITE SEMIFIELDS 3

Now we show that S(U, W ) is a semifield spread. Let {ei | i = 1, . . . , n} be a basis for n F viewed as a vector space over F0 and put e1 = 1. Let ψ be the bijection from F to F0 defined by n X ψ( aiei) = (a1, a2, . . . , an). i=1 Let V be a vector space of rank r + 1 over F and extend ψ in the natural way to a bijection between V and V1 ⊕ Vr. Let {fi | i = 1, . . . , r + 1} be a basis for V over F, such that V1 is the image of hf1i under ψ and Vr is the image of hf2, . . . , fr+1i, and let r+1 α = (1, α2, . . . , αr+1) ∈ F . Consider the element φα of GL(V ) defined by φα(f1) = α and φα(fi) = fi for i = 2, . . . , r + 1. The image of the vector (x1, . . . , xr+1) ∈ V is −1 (x1, x2 + α2x1, . . . xr+1 + αr+1x1). The composition ψα := ψφαψ induces a semilinear automorphism of V1 ⊕ Vr, moreover this automorphism will fix Vr pointwise and will fix A(U) := hU, (1, 0,..., 0)i setwise. Let S and T be distinct elements of B(A(U))\B(U) and −1 suppose S ∩ A(U) = hsi and T ∩ A(U) = hti. Then ψφαψ (t) = ψ(1, t2 + α2, . . . , tr+1 + αr+1), and choosing αk = sk − tk we obtain an element of GL(V1 ⊕ Vr) fixing A(U) and taking T to S. The set of collineations obtained in this way form a group of order qn acting transitively on the elements of B(hv, Ui) \ B(U) and fixing Vr pointwise. In the quotient geometry this group induces a group fixing Vr/W and acting transitively on the other elements of S(U, W ). Hence S(U, W ) is a semifield spread.

Let S(U, W ) denote the semiﬁeld corresponding to the semiﬁeld spread S(U, W ).

3. The construction covers all finite semifields

Let S = (F, +, ◦) be a finite semifield of order qn. With S there is associated a cubical array (see Knuth [7]), and without loss of generality we may assume that the entries of the associated cubical array are elements of F0. Define

Sx := {(y, y ◦ x) | y ∈ F}, for every x ∈ F and S∞ := {(0, y) | y ∈ F}. Then Σ(S) := {Sx | x ∈ F} ∪ {S∞} is a spread of the vector space V (F × F) of rank 2n over F0 consisting of subspaces of rank n over F0. This spread is a semifield spread with respect to S∞ (see [3]) and we call Σ(S) the semifield spread associated with S. The following theorem shows that the construction we explained in the previous section covers all finite semifields.

Theorem 3.1. For every finite semifield S, there exist subspaces U and W such that the semifield spread associated with S is isomorphic to S(U, W ).

Proof. Following [9] we may assume that S = (F, +, ◦) and multiplication in S is given by n−1 n−1 X qi qj X qj y ◦ x = cijx y = cj(x)y i,j=0 j=0

for some cij ∈ F and x ◦ y = 0 implies that x = 0 or y = 0. 4 SIMEON BALL, GARY EBERT, AND MICHEL LAVRAUW

Consider V1 ⊕ Vn as a vector space of rank n + 1 over F. The spread element of the n+1 Desarguesian spread S(V1 ⊕ Vn) containing x ∈ F is the rank n subspace over F0

{(yx0, yx1, . . . , yxn) | y ∈ F}.

Let U and W be the subspaces of Vn defined by q−1 q−n+1 U = {(0, c0(x), c1(x) , . . . , cn−1(x) ) | x ∈ F} and let n−1 X qi W = {(0, − zi , z1, z2, . . . , zn−1) | zi ∈ F}. i=1 If B(U) ∩ B(W ) 6= ∅ then there is an element of the Desarguesian spread meeting both U and W and so there is a n−1 X qi z = (0, − zi , z1, z2, . . . , zn−1) ∈ W i=1 and a y ∈ F∗ with the property that y−1z ∈ U. Hence there exists an x ∈ F∗ such that q−i zi = yci(x) , i 6= 0 and n−1 X qi − zi = yc0(x). i=1 Substituting for the zi in the second equality we get n−1 X qj y ◦ x = cj(x)y = 0 j=0 which implies x = 0 or y = 0, a contradiction. We have shown that B(U) ∩ B(W ) = ∅. Let v = (1, 0,..., 0) ∈ V1. The element of the Desarguesian spread S(V1 ⊕ Vn) containing the vector q−1 q−n+1 (1, c0(x), c1(x) , . . . , cn−1(x) ) ∈ hU, vi is q−1 q−n+1 Rx := {(y, yc0(x), yc1(x) , . . . , ycn−1(x) ) | y ∈ F}. Remark that the quotient space (V1 ⊕ Vn)/W is isomorphic to the projection of all subspaces containing W onto a subspace of rank 2n intersecting W trivially. In order to calculate the quotient subspace hRx,W i/W we first form the span n−1 X qi q−1 q−n+1 hRx,W i = {(y, yc0(x) − zi , yc1(x) + z1, . . . , ycn−1(x) + zn−1) | y, zi ∈ F}, i=1 and then intersect this with the subspace defined by X2 = X3 = ... = Xn = 0. Hence n−1 X qj hRx,W i/W = {(y, cj(x)y ) | y ∈ F} = {(y, y ◦ x | y ∈ F}. j=0 This shows that the semifield spread S(U, W ) is isomorphic to the semifield spread associated with S. A NEW APPROACH TO FINITE SEMIFIELDS 5

This shows that any semiﬁeld can be constructed for r = n, and thus conceivably for a smaller integer r.

4. Semifield operations

In [7] Knuth noted the group S3 acts on the set of finite semifields. The group S3 is gener- ated by the involutions τ1, which changes the order of multiplication and is equivalent to dualising the semifield plane, and τ2 which dualises the semifield spread; this geometrical interpretation was first observed in [2] and elaborated in [6]. Let S(F, ◦, +) be a semifield with multiplication defined by n−1 X qi qj y ◦ x = cijx y , i,j=0 for some cij ∈ F. For legibility reasons we will sometimes write ci,j instead of cij. Clearly

τ1(cij) = cji.

Let T ⊥ denote the dual of subspace T with respect to the duality corresponding to the form ((y, z), (u, v)) = T r(yv − zu),

where T r is the trace function from F to F0. Dualising the spread element Sx we get n−1 ⊥ X qi qj Sx = {(u, v) | T r(yv − cijx y u) = 0 ∀y ∈ F}. i,j=0 Now n−1 n−1 n−1 X qi qj X q−j q−j qi−j X qj qj qi T r(yv − cijx y u) = T r(y(v − cij u x )) = T r(y(v − ci−j,−ju x )), i,j=0 i,j=0 i,j=0

where the subscripts of ci,j are taken modulo n. Since T r(yλ) = 0 ∀y ∈ F implies λ = 0, we obtain n−1 ⊥ X qj qj qi Sx = {(u, ci−j,−ju x ) | u ∈ F}, i,j=0 and we may conclude that qj τ2(cij) = ci−j,−j.

The two operations τ1 and τ2 generate the six semifields related by the Knuth operations. The corresponding multiplications are determined by qj qi ci,j, τ1(cij) = cji, τ2(cij) = ci−j,−j, τ2τ1(cij) = cj−i,−i, qj qi τ1τ2(cij) = c−j,i−j, τ1τ2τ1(cij) = τ2τ1τ2(cij) = c−i,j−i. However in the case when U and W are both rank n subspaces (r = 2) we can reverse the roles of U and W and we can construct the semifield spread S(W, U). We now determine the effect of this operation, which we define as τ3, on the multiplication of the semifield S. Straightforward counting shows that there is always an element of the Desarguesian spread skew from U and W and without loss of generality we may assume that {(0, 0, y) | y ∈ F} 6 SIMEON BALL, GARY EBERT, AND MICHEL LAVRAUW

Pn−1 qi satisﬁes this condition. Therefore there are functions f(x) = i=0 fix and g(x) = Pn−1 qi i=0 gix such that U = {(0, x, f(x)) | x ∈ F} and W = {(0, z, g(z)) | z ∈ F}. Let v = (1, 0, 0), so that the spread elements belonging to B(hv, Ui) \ B(U) are

Rx := {(y, yx, yf(x)) | y ∈ F}, x ∈ F. The span hRx,W i = {(y, yx + z, yf(x) + g(z)) | y, z ∈ F} meets X2 = 0 when z = −xy and so in the quotient space of W

hRx,W i/W = {(y, yf(x) − g(xy)) | y ∈ F}. The multiplication ◦ in the corresponding semiﬁeld is given by n−1 X qi qi y ◦ x = yf(x) − g(xy) = (fiyx − gi(xy) ). i=0 Therefore cij = δ0jfi − δijgi,

where δij is the Kronecker delta. Thus the semifield S(W, U) obtained by reversing the roles of U and W has multiplication defined by −ci,i−j = − δ0(i−j)fi + δi(i−j)gi = −δijfi + δ0jgi. Hence in addition to the six semifields which have multiplication given by

cij, τ1(cij), τ2(cij), τ2τ1(cij), τ1τ2(cij), τ1τ2τ1(cij) and are related by the Knuth operations, we have another six semiﬁelds which have multiplication determined by

τ3(cij), τ1τ3(cij), τ2τ3(cij), τ2τ1τ3(cij), τ1τ2τ3(cij), τ1τ2τ1τ3(cij), which are qi−j qi qi−j qi −ci,i−j, −ci−j,i, −cj,j−i, −c−j,−i, −cj−i,j, −c−i,−j, respectively.

Now what we would like to do is to apply τ3 to these semifields, however we have only shown that we can do so in the case that they arise from the construction with r = 2, i.e., from two subspaces of rank n in a vector space of rank 2n. Obviously this is the 2 case for τ3(ci,j) since it is associated to the semifield spread S(W, U), but τ3 = 1 and we get back to the six semifields we had before. The only other semifield we know to be a semifield arising from two subspaces of rank n in a vector space of rank 2n that we have not yet considered is the semifield constructed from S(W ⊥,U ⊥). With the same notation as before it is a straightforward calculation to obtain U ⊥ = {(0, −fˆ(b), b) | b ∈ F} and ⊥ ˆ Pn−1 1/qi Pn−1 1/qi W = {0, −gˆ(d), d) | d ∈ F}, where f(x) = i=0 (fix) andg ˆ(x) = i=0 (gix) . Applying the construction one obtains the multiplication y ◦ x = fˆ(xy) − ygˆ(x), A NEW APPROACH TO FINITE SEMIFIELDS 7

qi qj qi and as before we obtain that the new coefficient of the term x y is c−i,j−i, which is the same as for the semifield obtained by applying τ1τ2τ1 to S(U, W ), and so this will generate no further semifields. We conclude that a semifield arising from two subspaces of rank n in a vector space of rank 2n generates at least twelve (not necessarily mutually non-isotopic) semifields.

Remark 4.1. We remark that the operation τ2τ1τ2τ3 is a generalisation of the the geometric operation S 7→ Sˆ as explained in [8] in the case of semifields corresponding to a semifield flock.

References [1] A. A. Albert, Finite division algebras and finite planes, Proc. Sympos. Appl. Math., Vol. 10, 53–70, American Mathematical Society, Providence, R.I., 1960. [2] S. Ball and M. R. Brown, The six semifield planes associated with a semifield flock, Adv. Math., 189 (2004) 68–87. [3] P. Dembowski, Finite Geometries, Springer, Berlin, 1968. [4] L. E. Dickson, On finite algebras, GöttingenNachrichtung, (1905) 358–393. [5] D. R. Hughes and F. C. Piper Projective Planes. Springer, Berlin, 1973. [6] W. M. Kantor, Commutative semifields and symplectic spreads, J. Algebra, 270 (2003) 96–114. [7] D. E. Knuth, Finite semifields and projective planes, J. Algebra, 2 (1965) 182–217. [8] M. Lavrauw, The two sets of three semifields associated with a semifield flock. preprint. [9] G. Menichetti n-dimensional algebras over a field with a cyclic extension of degree n, Geom. Dedicata 63 (1996) 69–94.