The first layer of \({\mathbb{Z}}^2_2\)-extension over imaginary quadratic fields (Q5928554)

Let \(k\) be a complex quadratic number field; it is known that, for every prime \(p\), \(k\) has two independent \(\mathbb Z_p\)-extensions whose layers are normal over \(\mathbb Q\): the cyclotomic \(\mathbb Z_2\)-extension, which is abelian over \(\mathbb Q\), and the anticyclotomic \(\mathbb Z_p\)-extension \(k_\infty/k\), for which the nontrivial automorphism of \(k/\mathbb Q\) acts as \(-1\) on \(\text{Gal}(k_\infty/k)\). NEWLINENEWLINENEWLINEIn this paper, the author studies the first layer \(k_1/k\) of the anticyclotomic \(\mathbb Z_2\)-extension of \(k\). Let \(F_1\) denote the compositum of the first layers of all \(\mathbb Z_2\)-extensions of \(k\); then \(F_1/\mathbb Q\) is normal with Galois group \(D_4\) (the dihedral group of order \(8\)) or \((\mathbb Z/2\mathbb Z)^3\). Theorem 1 claims that \(F_1/\mathbb Q\) has Galois group \(D_4\) if \(2\) splits in \(k\), and \((\mathbb Z/2\mathbb Z)^3\) if \(k = \mathbb Q(\sqrt{-d})\) with \(d \equiv 3, 5 \bmod 8\). In Theorems 4 and 5, \(F_1\) is given explicitly for the complex quadratic fields with class number \(1\). NEWLINENEWLINENEWLINEReviewer's remark: It seems that the proof of Theorem 1 is only correct if \(k\) has odd class number.