Showing papers on "Generic polynomial published in 1986"

Roots of Affine Polynomials

Abstract: Publisher Summary A Galois field F = GF(q) of order q = p h is considered in the chapter, where p is a prime, and K = GF(q n ) is considered an algebraic extension of a given degree n > 1. An affine polynomial—of K[x]over F—is a polynomial that can be expressed as deference between L(x) and b, where b belongs to K. Hence, the determination of (eventual) roots of the polynomial in K can be reduced to the determination of solutions of a linear system of equations in indeterminates x i and with coefficients in F. This procedure is, however, tedious also in the most simply cases and does not decise a priori how many roots in K exist. The conditional equation for affine polynomial has roots in K only if the certain system of linear equations has solutions. The roots of the equation are expressible as functions of certain solutions of the linear system. In particular, the obtained results are useful also in the case where the coefficients of the affine polynomial are not constant.

AT2-Optimal Galois Field Multiplier for VLSI

Abstract: For every prime p, there are AT2-optimal VLSI multipliers for Galois fields GF(pn) in standard notation. In fact, the lower bound AT2 = Ω(n2) is matched for every computation time T in the range [Ω(log n), 0(√n)]. Similar results hold for variable primes p too. The designs are based on the DFT on a structure similar to Fermat rings. For p=2 the DFT uses 3l-th instead of 2l-th rotts of unity.

Genus group of finite Galois extensions

Abstract: Let K/k be a Galois extension of finite degree, and let K' denote the maximal abelian extension over k contained in the Hilbert class field of K. We give formulas about the group structure of Gal(K'/k) and the genus group of K/k, which refine the ordinary genus formula. For an algebraic number field k of finite degree, let Cl(k) denote the ideal class group of k. For a modulus S of k (i.e., a finite product of primes of k), let Ik(S), Pk(S), and PkS denote the group of ideals of k prime to S, the group of principal ideals of k prime to S, and the ray ideal group modulo S in k, respectively. Similarly, let k(S) and ks denote the group of elements of k prime to S and the ray number group of k modulo S, respectively. Let K/k be a Galois extension of finite degree. Let K' be the maximal abelian extension over k contained in the Hilbert class field K of K. Then, by definition, the genus field K* of K/k in the wide sense is KK', and the genus group of K/k is Gal(K*/K). The following lemma is well known and proved by a standard manner in class field theory. LEMMA 1. Let f' be the conductor of K'/k; then K' is a class field over k corresponding to Ik( f')/NK/k(PK( 0'))Pkf,. Moreover, if K/k is abelian, then f' coincides with the conductor of K/k. For a modulus f with f'l f, put 05(K/k) = Ik(f)/NK/k(PK(M))Pkf. Clearly, S(K/k) does not depend on the choice of such f (up to isomorphisms). The purpose of this paper is to describe the l'-rank of 5 (K/k). Let / be a prime number. Throughout this paper we fix / unless otherwise stated. For an abelian group A written multiplicatively, let ranki(A) denote the l'-rank of A, i.e., the F,-dimension of A"'/A". For i > 0, put Fi = {a E kX l(a) E P,)}. Then kxD F1 DF2D Fi D Ek and Fi D Fi_lEk, where Ek denotes the group of units in k. Put Fi(S) = Fi n k(S). LEMMA 2. Let 1 -+ N -+ M -+ L --+1 (N c M) be an exact sequence of finite abelian groups. Put Ni = N n M". Then for i > 1, we have rank,(M) = rank,(L) + rank1(NiA1/NA) = ranki(L) + logif #(N,-11NJ)} Received by the editors October 11, 1985. 1980 Mathematics Subject Classification (1985 Revision). Primary 1IR37, 1IR65. ?)1986 American Mathematical Society 0002-9939/86 $1.00 + $.25 per page