Showing papers on "Generic polynomial published in 1970"

On Artin's L-Functions

Abstract: The nonabelian Artin L-functions and their generalizations by Weil are known to be meromorphic in the whole complex plane and to satisfy a functional equation similar to that of the zeta function but little is known about their poles. It seems to be expected that the L-functions associated to irreducible representations of the Galois group or another closely related group, the Weil group, will be entire. One way that has been suggested to show this is to show that the Artin L-functions are equal to L-functions associated to automorphic forms. This is a more difficult problem than we can consider at present. In certain cases the converse question is much easier. It is possible to show that if the L-functions associated to irreducible two-dimensional representations of the Weil group are all entire then these L-functions are equal to certain L-functions associated to automorphic forms on GL(2). Before formulating this result precisely we review Weil’s generalization of Artin’s L-functions. A global field will be just an algebraic number field of finite degree over the rationals or a function field in one variable over a finite field. A local field is the completion of a global field at some place. If F is a local field let CF be the multiplicative group of F and if F is a global field let CF be the id` ele class group of F .I fK is a finite Galois extension of F the Weil group WK/F is an extension ofG(K/F ), the Galois group of K/F ,b yCK. Thus there is an exact sequence

Inseparable Galois theory of exponent one

Abstract: An exponent one inseparable Galois theory for commutative ring extensions of prime characteristic p ?0 is given in this paper. Let C be a commutative ring of prime characteristic p $0. Let g be both a C-module and a restricted Lie ring of derivations on C and denote by A the kernel of g, i.e., the set of all x in C such that t3x=0 for all a in g. We say C over A is a purely inseparable Galois extension of exponent one if and only if C is finitely generated projective as A-module and C[g] = HomA (C, C). In this paper, we present a Galois correspondence between the restricted Lie subrings of g which are also C-module direct summands of g and the intermediate rings between C and A over which locally C admits p-basis. The Galois hypothesis C[g]= HomA (C, C) used here is an analog of the separable Galois hypothesis used in [7] and [8]. In case C is a field, our theory reduces to Jacobson's Galois theory for purely inseparable field extensions of exponent one. In a subsequent paper [6], we shall present the attendant Galois cohomology results. Among other things, we shall show that there is an exact sequence 0 -O L(C/A) -P(A) -P(C) -&(g, C) -B(C/A) -O 0, where B(C/A) is the Brauer group for C over A, 9(g, C) is Hochschild's group of regular restricted Lie algebra extensions of C by g, P is the functor of taking rank one projective class group and L(C/A) is the logarithmic derivative group. We also show that the Amitsur cohomology groups Hn + 2(C/A, Gm), n >0, are isomorphic to Hochschild's groups (Cn OA g, Cn +1) of regular restricted Lie algebra extensions of Cn + 1, the n + 1-fold tensor product C OA ... OA C, by Cn ?A gAll rings in the following are assumed to be commutative with 1. If A is a subring of a ring C we understand that both A and C have the same identity. By an A-algebra C we mean that A is a subring of C. Finally all ring-homomorphisms and modules are unitary. 1. LEMMA. Let C be a ring of prime characteristic p & 0, and let A be a subring of C such that tP E A for all t in C. Then Spec C is canonically homeomorphic to Spec A. Received by the editors July 8, 1968 and, in revised form, October 2, 1969. AMS Subject Classifications. Primary 1370.

The uses of computers in Galois theory

Abstract: Publisher Summary This chapter discusses the use of computers in Galois theory. The problem of calculating the Galois group of a polynomial over the rationals is remarkable among mathematical algorithms for the paucity of its input–output. A single polynomial is given as input, and a single group code or the Cayley table of a group is returned as output. The chapter describes the computational difficulties that arise in computing such groups. It has been noticed that although the splitting fields whose automorphism groups are the Galois groups of polynomials over the rationals are infinite fields, the problem of calculating these automorphism groups is actually a finite problem. Simpler methods are given by van der Waerden in the case in which the polynomial has degree less than or equal to 4.