Normal integral basis of an unramified quadratic extension over a cyclotomic \mathbb{Z}_2-extension

Abstract

Let ℓ be an odd prime number. Let K/ℚ be a real cyclic extension of degree ℓ, A K the 2-part of the ideal class group of K, and H/K the class field corresponding to A K /A K 2 . Let K n be the nth layer of the cyclotomic ℤ 2 -extension over K. We consider the questions (Q1) “does H/K has a normal integral basis?”, and (Q2) “if not, does the pushed-up extension HK n /K n has a normal integral basis for some n≥1?” Under some assumptions on ℓ and K, we answer these questions in terms of the 2-adic L-function associated to the base field K. We also give some numerical examples.