Showing papers on "Generic polynomial published in 1982"
TL;DR: In this paper, a useful criterion for characterizing a monic irreducible polynomial over Q with Galois group Dp (the dihedral group of order 2p, p: prime) is given by making use of the geometry of Dp, i.e., Dp is the symmetry group of the regular pgon.
30 citations
TL;DR: For a commutative ring R and a finite abelian group G the following conditions are equivalent: (a) Gal(R,G)=Gal (R[X],G), i.e. every commutive Galois extension of R[X] with Galois group G is extended from R.
Abstract: Our main result states that for a commutative ring R and a finite abelian group G the following conditions are equivalent: (a) Gal(R,G)=Gal (R[X],G), i.e. every commutative Galois extension of R[X]with Galois group G is extended from R. (b) The order of G is a non-zero-divisor in R/Nil(R). The proof uses lifting properties of Galois extensions over Hensel pairs and a “Milnor-type” patching theorem.
2 citations
TL;DR: In this article, Borevich showed that Λ-module E has a system of n+1 generators, of which n−1 are free and two are connected by certain relations.
Abstract: Suppose k is a local field that is an extension of the field of p -adic numbers of degree n and does not contain a primitive p -th root of 1, and suppose K/k is a cyclic p-extension with Galois group G. The group E of principal units of K is a multiplicatively written module over the group ring Λ=ℤp[G], where ℤp is the ring of p-adic integers. It was shown by Borevich (Ref. Zh. Mat., 1965, 3A256) that the Λ-module E has a system of n+1 generators, of which n−1 are free and two are connected by certain relations. In the present paper these Λ-generators are constructed explicitly and their arithmetical characteristics indicated.
1 citations
TL;DR: In this paper, the authors define the notion of nilpotency of elements in a polynomial algebra over a field of characteristic not 2 or 3 and prove that the algebra of polynomials in y over y is associative.