Kummer extensions with few roots of unity

Abstract

Let K be an arbitrary field, K an algebraic closure of K, n ≥ 1 a natural number, and μ n( K) = {z|z ∈ K,z n = 1} . A finite Kummer extension of K of exponent n with few (resp., many) roots of unity is an extension K( x 1,…, x k ) of K, where k ∈ N ∗ , x 1, …, x k ∈ K ∗ are such that x i n ∈ K for all i, 1 ≤ i ≤ k, and μ n( K) ∩ K(x 1, …, x k) ⊆ {1, −1} (resp., μ n( K) ⊆ K) . We prove that a classical result concerning the evaluation of the degree [ K( x 1, …, x k ): K] holds equally for finite Kummer extensions of exponent n with few or with many roots of unity, if Char(K) ∤ n . For such an extension K ⊆ K( x i , …, x k ) for which [ K( x 1, …, x k ) : K] = Π 1 ≤ i ≤ k [ K( x i ) : K], it is shown that K( x 1, …, x k ) = K( x 1 + ctdot; + x k ). Further, if K is an arbitrary field and n is a prime number other than Char( K), then any extension K ⊆ K( x 1, …, x k ), where k ∈ N ∗ and x 1, …, x k ∈ K ∗ are such that x i n ∈ K for all i, 1 ≤ i ≤ k, is a finite Kummer extension of exponent n with few or with many roots of unity, and, consequently, the above results hold in this case. Our results complete, unify, or extend some of the results of J. L. Mordell, H. Hasse, A. Baker and H. M. Stark, I. Kaplansky, I. Richards, and H. D. Ursell appearing in the literature, and reveal the connections between them.