Kummer Theory for Multiquadratic or Quartic Cyclic Number Fields

Abstract

Let K be a number field which is multiquadratic or quartic cyclic. We prove several results about the Kummer extensions of K, namely concerning the intersection between the Kummer extensions and the cyclotomic extensions of K. For G a finitely generated subgroup of K ×, we consider the cyclotomic-Kummer extensions K ( ζ n t , G n ) / K ( ζ n t ) K\left( {{\zeta _{nt}},\root n \of G } \right)/K\left( {{\zeta _{nt}}} \right) for all positive integers n and t, and we describe an explicit finite procedure to compute at once the degree of all these extensions.

Kummer theory for products of one-dimensional tori

“…) is generated by squareroots of elements of K(T [m]). Indeed, if P = (x, y) ∈ G i \ T i [2], then by [3, Lemma 3.1] we have K( 12 P ) = K( 2(x + 1)). Proof of Theorem 1.1.…”

“…We may suppose that the image of G is torsion free up to replacing m by lcm(m, nt), where t is the order of its torsion subgroup (notice that t | 24 because L is multiquadratic). Calling G ′ the image of this group in L × m , by [2] we may compute the degree of all extensions We now determine those m ⩾ 3 such that…”

“…It is possible to compute at once the degree of all number fields Q(T [m], 1 n G), where m, n are positive integers such that n divides m. The above result is stated over Q for simplicity, however one may generalize it to those number fields such that the analogous computations are feasible. For example, by the results in [2] we have the following: Remark 1.3. -In Theorem 1.1 we may compute at once the degree of the torsion-Kummer extensions for all m and n if the splitting field of T is multiquadratic.…”