On Double Covers of the Generalized Symmetrical Group Zd ≀ Sm as Galois Groups over Algebraic Number Fields K with μd ⊂ K

Galois Groups of Periodic Points

“…Condition iii does not hold over R, because the primes dividing ⌬ 5, 5 must ramify in K. However, there are prime divisors of K which divide ⌬ to just the first power. To see this, recall that the ramification indices 5,5 in ⌳rQ of prime divisors P of ⌬ are 2. Hence the inertia group 5,5 Ž .…”

“…To see this, recall that the ramification indices 5,5 in ⌳rQ of prime divisors P of ⌬ are 2. Hence the inertia group 5,5 Ž . Ä 4 G P s 1, , for an involution .…”

“…Ž that ⌽ x is absolutely irreducible over F for all odd primes p see also 5 p w x . 9, Proposition 18 .…”

“…It follows that the ramification index of ဧ in KrQ is 1, and therefore ဧ divides ⌬ to exactly the first power. 5,5 Now let S be the set of prime ideals of K which have ramification index 2 over Q and let R be the subring of K which consists of the S-integers S Ž .…”

“…then R is a Dedekind domain and conditions ii and iii hold for the S Ž . Ž polynomial ⌽ x and its discriminant factors ⌬ and ⌬ over R see 5 5 , 1 5 , 5 S . Example 1 in Section 3 .…”