William Yslas Vélez

The lattice of subfields of a radical extension

Abstract Let xm − a be irreducible over F with char F∤m and let α be a root of xm − a. The purpose of this paper is to study the lattice of subfields of F(α) F and to this end C( F(α) F , k) is defined to be the number of subfields of F(α) of degree k over F. C( F(α) F , p n ) is explicitly determined for p a prime and the following structure theorem for the lattice of subfields is proved. Let N be the maximal normal subfield of F(α) over F and set n = |N : F|, then C( F(α) F , k) = C( F(α) F , (k, n)) = C( N F , (k, n)) . The irreducible binomials xs − b, xs − c are said to be equivalent if there exist roots βs = b, γs = a such that F(β) = F(γ). All the mutually inequivalent binomials which have roots in F(α) are determined. Finally these results are applied to the study of normal binomials and those irreducible binomials x2e − a which are normal over F (char F ≠ 2) together with their Galois groups are characterized.

The Galois group of a radical extension of the rationals

The Galois group of the splitting field of an irreducible binomialx2e−a overQ is computed explicitly as a full subgroup of the holomorph of the cyclic group of order 2e. The general casexn−a is also effectively computed.

Uniform distribution of two-term recurrence sequences

Etude de la distribution uniforme modulo m des suites de recurrences a deux termes pour m=p k , p etant premier

The torsion group of a field defined by radicals

Abstract Let L F be a finite separable extension, L ∗ = L{0} , and T( L ∗ F ∗ ) the torsion subgroup of L ∗ F ∗ . When L F is an abelian extension T( L ∗ F ∗ ) is explicitly determined. This information is used to study the structure of T( L ∗ F ∗ ) . In particular, T( F(α) ∗ F ∗ ) when a m = a ∈ F is explicitly determined.

A note on the normality of unramified, abelian extensions of quadratic extensions

AbstractLet F, K and L be algebraic number fields such that

Resetting the bar for graduate admissions

F \subseteq K \subseteq L

On normal binomials

, [K∶F]=2 and [L∶K]=n. It is a simple consequence of the class field theory that, if L is an abelian, unramified extension of K and (n,h)=1, where h is the class number of F, then L is normal over F. The purpose of this note is to point out the necessity of the condition (n,h)=1 by constructing for any field F with even class number a tower of fields