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28
Normal Bases over Finite Fields
, 1993
"... Interest in normal bases over finite fields stems both from mathematical theory and practical applications. There has been a lot of literature dealing with various properties of normal bases (for finite fields and for Galois extension of arbitrary fields). The advantage of using normal bases to repr ..."
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Interest in normal bases over finite fields stems both from mathematical theory and practical applications. There has been a lot of literature dealing with various properties of normal bases (for finite fields and for Galois extension of arbitrary fields). The advantage of using normal bases to represent finite fields was noted by Hensel in 1888. With the introduction of optimal normal bases, large finite fields, that can be used in secure and e#cient implementation of several cryptosystems, have recently been realized in hardware. The present thesis studies various theoretical and practical aspects of normal bases in finite fields. We first give some characterizations of normal bases. Then by using linear algebra, we prove that F q n has a basis over F q such that any element in F q represented in this basis generates a normal basis if and only if some groups of coordinates are not simultaneously zero. We show how to construct an irreducible polynomial of degree 2 n with linearly i...
Splitting fields for E8torsors
 Department of Mathematics, University of Southern
"... Several of the fundamental problems of algebra can be unified into the problem of classifying Gtorsors over an arbitrary field k, for a linear algebraic group G. (A Gtorsor can be defined as a principal Gbundle over Spec k, or as an algebraic variety over k with a free transitive action of G.) Fo ..."
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Several of the fundamental problems of algebra can be unified into the problem of classifying Gtorsors over an arbitrary field k, for a linear algebraic group G. (A Gtorsor can be defined as a principal Gbundle over Spec k, or as an algebraic variety over k with a free transitive action of G.) For example, PGL(n)torsors are equivalent to central simple algebras, and torsors for the orthogonal group are equivalent to quadratic forms. See Serre [36] for a recent survey of the classification problem for Gtorsors over a field. The study of Gtorsors is still in its early stages. Indeed, it is not completely known how complicated Gtorsors can be, if we fix the type of the group but allow arbitrary base fields. Tits showed that there is a bound on how complicated they can be. For each split semisimple group G, there is an integer d(G) depending only on the type of G, not on the field, such that every Gtorsor over a field k becomes trivial over some finite extension E of k of degree dividing d(G) [40]. For example, it is easy to see that one can take d(G) to be the order of the Weyl group of G. But it is a fundamental problem to determine the best possible number d(G) for
HYPERBOLICITY OF ORTHOGONAL INVOLUTIONS
"... Abstract. We show that a nonhyperbolic orthogonal involution on a central simple algebra over a field of characteristic ̸ = 2 remains nonhyperbolic over some splitting field of the algebra. 1. ..."
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Abstract. We show that a nonhyperbolic orthogonal involution on a central simple algebra over a field of characteristic ̸ = 2 remains nonhyperbolic over some splitting field of the algebra. 1.
Structure theorems for Jordan algebras of degree three over fields of arbitrary characteristic
 Comm. Algebra
"... Classical results, like the construction of a 3fold Pfister form attached to any central simple associative algebra of degree 3 with involution of the second kind [HKRT], or the SkolemNoether theorem for Albert algebras and their 9dimensional separable subalgebras [PaST], which originally were de ..."
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Cited by 7 (2 self)
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Classical results, like the construction of a 3fold Pfister form attached to any central simple associative algebra of degree 3 with involution of the second kind [HKRT], or the SkolemNoether theorem for Albert algebras and their 9dimensional separable subalgebras [PaST], which originally were derived only over fields of characteristic not 2 (or 3), are extended here to base fields of arbitrary characteristic. The methods we use are quite different from the ones originally employed and, in many cases, lead to expanded versions of the aforementioned results that continue to be valid in any characteristic. Thanks to their close connection with the Galois cohomology of classical and exceptional groups, Jordan algebras of degree 3 have attracted considerable attention over the last couple of years. The results on the (cohomological) invariants mod 2, based to a large extent on the construction of HaileKnusRostTignol [HKRT] attaching a 3fold Pfister form to any central simple associative
Galois Algebras, Hasse Principle and Induction–Restriction Methods
"... Let k be a field of characteristic ̸ = 2, and let L be a Galois extension of k with group G. Let us denote by qL: L × L → k the trace form, defined by qL(x,y) = Tr L/k(xy). Let (gx)g∈G be a normal basis of L over k. We say that this is a self–dual normal basis if qL(gx,hx) = δg,h. If the order of ..."
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Let k be a field of characteristic ̸ = 2, and let L be a Galois extension of k with group G. Let us denote by qL: L × L → k the trace form, defined by qL(x,y) = Tr L/k(xy). Let (gx)g∈G be a normal basis of L over k. We say that this is a self–dual normal basis if qL(gx,hx) = δg,h. If the order of G is odd, then L always has a self–dual normal basis
METABOLIC INVOLUTIONS
"... Abstract. In this paper we study the conditions under which an involution becomes metabolic over a quadratic field extension. We characterise those involutions that become metabolic over a given separable quadratic extension. We further give an example of an anisotropic orthogonal involution that be ..."
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Abstract. In this paper we study the conditions under which an involution becomes metabolic over a quadratic field extension. We characterise those involutions that become metabolic over a given separable quadratic extension. We further give an example of an anisotropic orthogonal involution that becomes isotropic over a separable quadratic extension. 1.
Trace forms of GGalois algebras in virtual cohomological dimension 1 and 2
 Pacific J. Math
"... Let G be a finite group and let k be a field of char(k) ̸ = 2. We explicitly describe the set of trace forms of GGalois algebras over k when the virtual 2cohomological dimension vcd2(k) of k is at most 1. For fields with vcd2(k) ≤ 2 we give a cohomological criterion for the orthogonal sum of a tr ..."
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Let G be a finite group and let k be a field of char(k) ̸ = 2. We explicitly describe the set of trace forms of GGalois algebras over k when the virtual 2cohomological dimension vcd2(k) of k is at most 1. For fields with vcd2(k) ≤ 2 we give a cohomological criterion for the orthogonal sum of a trace form of a GGalois algebra with itself to be isomorphic to another such form.
Serre’s conjecture II for classical groups over imperfect fields
 J. of Pure and Applied Algebra
"... In his book “Cohomologie galoisienne, ” Serre formulates the following conjecture: Conjecture II: ([22, §3.1]) For every simply connected semisimple linear algebraic group G defined over a perfect field F of cohomological dimension at most 2, the Galois cohomology set H1(F,G) is trivial. ..."
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In his book “Cohomologie galoisienne, ” Serre formulates the following conjecture: Conjecture II: ([22, §3.1]) For every simply connected semisimple linear algebraic group G defined over a perfect field F of cohomological dimension at most 2, the Galois cohomology set H1(F,G) is trivial.
Sesquilinear forms over rings with involution
, 2012
"... Abstract Many classical results concerning quadratic forms have been extended to hermitian forms over algebras with involution. However, not much is known in the case of sesquilinear forms without any symmetry property. The present paper will establish a Witt cancellation result, an analogueof Sprin ..."
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Abstract Many classical results concerning quadratic forms have been extended to hermitian forms over algebras with involution. However, not much is known in the case of sesquilinear forms without any symmetry property. The present paper will establish a Witt cancellation result, an analogueof Springer’s theorem, as well as some localglobal and finiteness results in this context.