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A Survey of Finite Semifields Discrete Mathematics 208/209 (1999) 125–137 www.elsevier.com/locate/disc View metadata, citation and similar papers at core.ac.uk brought to you by CORE A survey of ÿnite semiÿelds provided by Elsevier - Publisher Connector M. Corderoa;∗, G.P. Weneb aDepartment of Mathematics, Texas Tech University, Lubbock, TX 79409, USA bDivision of Mathematics and Statistics, The University of Texas at San Antonio, San Antonio, TX 78249, USA Received 5 March 1997; revised 3 September 1997; accepted 13 April 1998 Abstract The ÿrst part of this paper is a discussion of the known constructions that lead to inÿnite families of semiÿelds of ÿnite order.The second part is a catalog of the known semiÿelds of ÿnite order. The bibliography includes some references that concern the subject but are not explicitly cited. c 1999 Elsevier Science B.V. All rights reserved. MSC: primary 17A35; secondary 12K10; 20N5 Keywords: Semiÿelds; Division rings; Finite geometries 1. Introduction A ÿnite semiÿeld is a ÿnite algebraic system containing at least two distinguished elements 0 and e. A semiÿeld S possesses two binary operations, addition and multi- plication, written in the usual notation and satisfying the following axioms: (1) (S; +) is a group with identity 0. (2) If a and b are elements of S and ab = 0 then a =0 or b=0. (3) If a, b and c are elements of S then a(b + c)=ab + ac and (a + b)c = ac + bc. (4) The element e satisÿes the relationship ea = ae = a for all a in S. All semiÿelds will be ÿnite unless it is speciÿcally stated otherwise. It is seen that the addition is commutative and that there are unique solutions to the equations ax = b and xa = b for every nonzero a and every b in S. Furthermore, S has a vector space structure over some prime ÿeld F = GF(p) and that S has pn elements where n is the dimension of S over F. ∗ Corresponding author. E-mail address: [email protected] (M. Cordero) 0012-365X/99/$ - see front matter c 1999 Elsevier Science B.V. All rights reserved. PII: S0012-365X(99)00068-0 126 M. Cordero, G.P. Wene / Discrete Mathematics 208/209 (1999) 125{137 The term semiÿeld is due to Knuth [77]; semiÿelds are also called division rings and division algebras (some authors do not insist that a division ring has a unity element). Any ÿnite ÿeld is an associative semiÿeld; the term proper semiÿeld will mean a semiÿeld in which the multiplication is not associative. Semiÿelds are also discussed in the texts of Dembowski [25], Hughes and Piper [52] and Stevenson [93]. The ÿrst part of this survey presents constructions leading to inÿnite families of semiÿelds. The second part catalogs individual semiÿelds by giving explicit construction and discussing the algebraic and geometric properties of each. 2. Inÿnite families Any semiÿeld of order p or p2 must be a ÿnite ÿeld. Knuth [77] proved that a proper semiÿeld of least order has 16 elements by showing that the only semiÿeld of order 23 is the ÿnite ÿeld GF(23). 2.1. The early work of Dickson The study of three-dimensional nonassociative division algebras began with Dickson [30] in 1905. He continued this study in [31,33,34] and gave a capsule summary in his monograph [35]. The ‘Dickson Commutative Division Algebras’ were introduced in the paper [32] and are of even dimension. The study of three-dimensional nonassociative division algebras was continued by Zelinsky [100] in his Master’s thesis. Menichetti [85,86] proved that any three- dimensional nonassociative division algebra was a twisted ÿeld (see below) and derived formulae for counting the number of these algebras [85]: (q3 − q2 + q − 10)=3ifq≡1 (mod 3); V = (q3 − q2 + q − 6)=3 otherwise; where q is the number of elements in the ÿnite ÿeld. Kaplansky [73,74] applied Dickson’s construction over algebraically closed ÿelds and Harrison [46] studies commutative algebras three-dimensional over an algebraically closed ÿeld of characteristic zero. Wene [99] began a study of semiÿelds three- dimensional over algebraic number ÿelds. The Dickson commutative semiÿelds are exhaustedly treated in the two papers by Burmester [12,13]. Let F be the ÿnite ÿeld GF(pn) where p is an odd prime and n¿1 and let f be any element of F that is not a square. The semiÿeld S has the structure of a two-dimensional vector space over F with basis elements 1 and .IfÂis an automorphism of F given by x = xpr ; 16r¡n; we can deÿne a multiplication in S by (a + b)(c + d)=(ac + f(bd)Â)+(ad + bc): The middle nucleus is the ÿeld F; the left and right nucleus is the subÿeld of F ÿxed by Â. M. Cordero, G.P. Wene / Discrete Mathematics 208/209 (1999) 125{137 127 These semiÿelds are discussed brie y on p. 195 in the text by Hughes an Piper [52]. 2.2. Zemmer's cyclic commutative semiÿelds Zemmer in his 1952 paper [101] investigates semiÿelds with predetermined au- tomorphism group. He proved the following: Let n be a positive integer and F = GF(qk );q¿2;a Galois ÿeld with qk =2ns +1; for some positive integer s. Then there exists a commutative, proper semiÿeld A∗ of dimension 2n over F with the following properties: (i) A∗ has a cyclic basis relative to F. (ii) The automorphism group of A∗ relative to F is the cyclic group of order 2n. (iii) A∗ contains a unique associative subalgebra of dimension n over F isomorphic to the ÿeld GF(qnk ). 2.3. The twisted ÿelds of Albert The twisted ÿelds were introduced by Albert [2] in 1952 who continued to study these semiÿelds in a series of papers [3–7]. Let F be the ÿnite ÿeld GF(pn) with n¿2. Let  and be automorphisms of F and let F and F be the subÿelds of F ÿxed by  and , respectively. The semiÿeld S has the structure of a vector space F over F ∩ F. Let e denote the unit element of F and let c be any element of F such that x y c 6= for any x; y ∈ F \{0}: x y Deÿne linear transformations Ae and Be on F by  xAe = x − cx and yBe = y − cy ; respectively:  The product ∗ in S will be given by xAe ∗ yBe = xy − cx y . The vector space S becomes a semiÿeld with unit element e − c (see [6] for details). Kaplansky [74] conjectured: Any three-dimensional semiÿeld over a ÿnite ÿeld is associative or a twisted ÿeld. Menichetti proved this conjecture true in his papers [85,86]. 2.4. Knuth's binary semiÿelds These commutative semiÿelds were constructed by Knuth [78] in 1965 from the ÿeld mn F = GF(2 ), where n is odd, n¿1. We follow Knuth’s construction. Let F0 be the m subÿeld GF(2 ). Considering F as a vector space over F0, let f be any nonzero linear functional from F to F0, i.e., f(a + b)=f(a)+f(b); 128 M. Cordero, G.P. Wene / Discrete Mathematics 208/209 (1999) 125{137 for all a; b ∈ F and all ; ∈ F0. Deÿne a multiplication in F as follows: a ◦ b = ab +(f(a)b+f(b)a)2: Now we can deÿne a product ∗ in the vector space F by (1 ◦ a) ∗ (1 ◦ b)=a◦b: The vector space becomes a semiÿeld. Knuth then computes all the automorphisms and autotopisms of this semiÿeld, as well as the collineation group of the corresponding plane. 2.5. Semiÿelds quadratic over a weak nucleus A subÿeld K of a semiÿeld S is called a weak nucleus if (xy)z −x(yz)=0 whenever two of x; y and z are in K. The semiÿeld S will be a vector space over it’s weak nucleus but right and left multiplication may not be linear transformations over the weak nucleus. If the dimension of S as a vector space over it’s weak nucleus is two, we say that S is quadratic over a weak nucleus. Knuth [77] (1965) gave conditions to determine all isotopism classes of semiÿelds quadratic over a weak nucleus. The constructions begin with a ÿnite ÿeld F = GF(pn);n¿2. In all cases the semi- ÿeld S has the structure of a two-dimensional vector space over a ÿeld F with basis elements 1 and . The Case I semiÿelds are also called generalized Dickson semiÿelds. The construction requires that the characteristic not be two and that the automorphisms , ÿ and are not all the identity automorphism. If the element f ∈ F is a nonsquare in F, the multiplication is given by (a + b)(c + d)=(ac + b dÿf)+(ad+bc): Dickson’s commutative semiÿelds form a special case and are the only commutative semiÿelds of this type. Let F be the ÿnite ÿeld GF(pn) with automorphism 6= I and let f; g be any elements of F such that y+1 + gy − f 6= 0 for y ∈ F: The multiplication of a Case II semiÿeld is given by one of the following, where the product (a + b)(c + d) is given by i. (ac + bd2 f)+(bc + ad + bdg) ii. (ac + bdf)+(bc + ad + bdg) iii. (ac + bd2 f)+(bc + ad + bdg) iv. (ac + bdf)+(bc + ad + bdg) and =−1. The semiÿelds constructed by Hughes and Kleinfeld [49] are a special case of this construction. The collineation group of the corresponding plane is computed in Hughes [50]. M. Cordero, G.P. Wene / Discrete Mathematics 208/209 (1999) 125{137 129 The concept of a weak nucleus is further exploited by Ganley [42] (1981) using the results in Cohen and Ganley [24]; new semiÿelds are discovered.
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