HILBERT-SPEISER NUMBER FIELDS AND STICKELBERGER IDEALS; THE CASE p = 2

We say that a number field F satisfies the condition (H′_2^m) when any abelian extension of exponent dividing 2^m has a normal basis with respect to rings of 2-integers. We say that it satisfies (H′ _2^∞) when it satisfies (H′ _2^m) for all m. We give a condition for F to satisfy (H'_2^m), and show that the imaginary quadratic fields F = Q(√−1) and Q(√−2) satisfy the very strong condition (H′ _2^∞) if the conjecture that h⁺_2^m = 1 for all m is valid. Here, h⁺_2^m) is the class number of the maximal real abelian field of conductor 2^m.