Showing papers on "Generic polynomial published in 1979"

Sequential Galois multiplication in GF(2n) with GF(2m) Galois multiplication gates

Abstract: Configurations of Boolean elements for implementing a sequential GF(2n) Galois multiplication gate are disclosed. Each configuration includes a single subfield GF(2m) Galois multiplication gate, where m is a positive integral divisor of n, e.g., n=8 and m=2, and assorted controls. Also disclosed is a sequential implementation of a GF(2n) Galois linear module as described in the J. T. Ellison Pat. No. 3,805,037 wherein the controls of the sequential GF(2n) multiply gate cause the Galois addition (bit-wise Exclusive-OR) of an n-bit binary vector, Z, to the final Galois product.

Local trace forms

Abstract: Let F be a local field of characteristic ≠2 and K a Galois extension field of F of degree n. Then K can be viewed as a quadratic space over F under the bilinear form T(x y)=trK/F xy for x y∊K. The invariants of this form are given in the case when n is odd; when n is even and F is nondyadic; and when n is evesF dyadic, and K/F is unramifed.

On Galois groups of class two extensions over the rational number field

Abstract: Let Q be the rational number field, K/Q be a maximal Abelian extension whose degree is some power of a prime l , and let f ( K ) be the conductor of K/Q ; if l = 2, let K be complex, and if in addition f ( K) ≡ 0 (mod 2), let f ( K) ≡ 0 (mod 16). Denote by ( K ) the Geschlechtermodul of K over Q and by K the maximal central l -extension of K/Q contained in the ray class field mod ( K ) of K . A. Frohlich [1, Theorem 4] completely determined the Galois group of K over Q in purely rational terms. The proof is based on [1, Theorem 3], though he did not write the proof in the case f ( K) ≡ 0 (mod 16). Moreover he gave a classification theory of all class two extensions over Q whose degree is a power of l . Hence we know the set of fields of nilpotency class two over Q , because a finite nilpotent group is a direct product of all its Sylow subgroups. But the theory becomes cumbersome, and it is desirable to reconstruct a more elementary one.

Pencils of higher derivations of arbitrary field extensions

Abstract: Let L be a field of characteristic p ^ 0. A subfield K of L is Galois if A' is the field of constants of a group of pencils of higher derivations on L. Let F d K be Galois subfields of L. Then the group of L over F is a normal subgroup of the group of L over K if and only if F = K(W') for some nonnegative integer r. If L/K splits as the tensor product of a purely inseparable extension and a separable extension, then the algebraic closure of X_in L, K, is also Galois in L. Given K, for every Galois extension Loi K, K is also Galois in L if and only if (K : Kp) < oo. 0. Introduction. Throughout we assume L is a field of characteristic p =£ 0. A rank / higher derivation on L is a sequence d = {di\0 < / < t + 1} of additive maps of L into L such that dr(ab)=^{dl(a)dJ(b)\i+j = r) and d0 is the identity map. The set of all rank / higher derivations forms a group with respect to the composition d ° e = /where fj = ~2{dmen\m + n = j}. Let H (L/K) be the set of all higher derivations on L trivial on K and having rank some power of p. Given d in H (L/K), v(d) = /where rank/ = p(rank d), f i = di and jC = 0 if p \j. Two higher derivations / and g are equivalent if g = v'(f) or/ = v'(g) for some i. The equivalence class of d is J and is called the pencil of d. The set of all pencils, H (L/K), can be given a group structure by defining df to be the pencil of d'f where d'Ed,f'Ef and rank d' = rank/' (3). A subfield K of L will be called Galois if K is the field of constants of a group of pencils on L or equivalently if L/K is modular and n,AT(Z/') = K (2, Proposition 1). In §1 it is shown that if F D K are Galois subfields of L, then H(L/F) is an invariant subgroup of H(L/K) if and only if F = K(LP) for some nonnegative integer r. This generalizes the result given in (2, Theorem 8) for the bounded exponent finite transcendence degree case. Let K denote the algebraic closure of K in L. L/K is said to split when L = J ®K D where J/K is purely inseparable and D/K is separable. §2 examines the question of when K is Galois in L, given L/K is Galois. Sufficient conditions are shown to be the splitting of L/K. Moreover, for every Galois extension L of K, K is also Galois in L if and only if Presented to the Society, August 10, 1978 under the title On pencil Galois theory; received by