Addendum to: Reductions of algebraic integers [J. Number Theory 167 (2016) 259–283]

The degree of Kummer extensions of number fields

“…Indeed, if is odd or ζ 4 ∈ K it suffices to take the formula provided by Theorem 13 (because m ω without loss of generality, the case m = 0 being trivial). Now suppose that = 2 and ζ 4 / ∈ K. If m 2, then we can extend the base field to K(ζ 4 ) and reduce to the previous case (notice that in [12] we proved that the 2-divisibility parameters of G over K(ζ 4 ) are determined by properties over K). We are left to compute the degree [K( √ G) : K], and this can be achieved with [4,Lemma 19].…”

“…Proof. Let s be as in (12). By [11,Theorem 2.7] there is a basis of G as a Z-module consisting of strongly -independent elements for all but finitely many primes , so that parameters for the -divisibility of G in K might be not all zero only for finitely many primes .…”

“…, n r , m with m max i (n i ). Indeed, by Proposition 20 we may replace each n i in (16) by min(n i , s), where s can be taken as in (12). Thus we may take A = rs , and hence by Proposition 17 we have A = 1 for all but finitely many prime numbers .…”

“…Notice that the bound A = rs is in general optimal. Indeed, for the case = 2 or ζ 4 ∈ K it suffices to consider s -th powers of strongly -independent elements of K, so that the parameters for the -divisibility in (12) are h i = 0 and d i = s for all i ∈ {1, . .…”

“…Consider the elements α 1 = 2, α 2 = 3, and α 3 = 5. The 3-divisibility parameters of the group G = 2, 3, 5 over Q are all zero, so by (12) we can take s = 0 in Lemma 15 and hence also in Propositions 18 and 20. We deduce that we have…”