Normal integral basis of an unramified quadratic extension over a cyclotomic ℤ₂-extension

Let ℓ be an odd prime number. LetK/ℚ be a real cyclic extension of degree ℓ,AK the 2-part of the ideal class group ofK, andH/Kthe class field corresponding to A K / A K 2 . LetKn be thenth layer of the cyclotomic ℤ₂-extension overK. We consider the questions (Q1) “doesH/Khas a normal integral basis?”, and (Q2) “if not, does the pushed-up extensionHKn/Kn has a normal integral basis for somen≥ 1?” Under some assumptions on ℓ andK, we answer these questions in terms of the 2-adicL-function associated to the base fieldK. We also give some numerical examples.