On the strongly ambiguous classes of some biquadratic number fields. (Q2828796)

In this paper the authors study the capitulation of \(2\)-ideal classes of imaginary biquadratic number fields of the form \(k=\mathbb{Q}(\sqrt{2pq},\text{i})\) where \(\text{i}=\sqrt{-1}\) and \(p\) and \(q\) are primes with \(p\equiv -q\equiv 1\mod{4}\). The authors claim to have proved that each strongly ambiguous ideal class of \(k/\mathbb{Q}(\text{i})\) capitulates in the absolute genus field \(k^*\) of \(k\), which generalizes a theorem of \textit{H. Furuya} [J. Number Theory 9, 4--15 (1977; Zbl 0347.12006)]. However, there is a problem in the authors' proof of this result, as follows. In Theorem 5.6: Part 2, with \(K_2=k(\sqrt{q})\) an unramified quadratic extension of \(k\), \(p=\pi_1\pi_2\), \(\pi_1=e+2\text{i}f\), \(\pi_2=e-2\text{i}f\), the fundamental unit of \(\mathbb{Q}(\sqrt{2pq})=x+y(\sqrt{2pq})\), and \(H_j\) and \(H_0\), respectively, being the prime ideal of \(k\) above \(\pi_j\) and \(1+i\), for \(j\) in \(\{ 1,2\}\), the authors claim that if the norm over the rational numbers \(\mathbb{Q}\) of the fundamental unit of \(\mathbb{Q}(\sqrt{2p})\) is equal to \(1\), and \(x+1\) and \(x-1\) are not squares in the natural numbers \(\mathbb{N}\), then there exists an unambiguous ideal \(I\) in \(k/\mathbb{Q}(i)\) of order \(2\) (this should be clarified that it is the ideal class \([I]\) that is of order \(2\)) such that the capitulation kernel from \(k\) to \(K_2\) is \(\langle[I]\rangle\) or \(\langle[H_0I]\rangle\) or \(\langle[H_1I]\rangle\) or \(\langle[H_0H_1I]\rangle\). But to prove this result, the authors claim that a particular unit in \(K_2\) is neither real nor purely imaginary, to obtain a contradiction. The problem is that there is an error in the proof of this claim, which one of the authors has acknowledged in a personal communication to me. Consequently the main results of this paper are substantially weakened, and the authors hope to be able to write a corrigendum to their paper to resolve the difficulties.NEWLINENEWLINE On a more minor note, there are a number of issues that should also be mentioned, as follows.NEWLINENEWLINE 1) The authors claim that a result of \textit{P. J. Sime} [Trans. Am. Math. Soc. 347, No. 12, 4855--4876 (1995; Zbl 0847.11060)] (which is stated as Lemma 4.4) along with a theorem by Chebotarev is sufficient to obtain a prime \(l\equiv 1\mod{4}\) such that the Kronecker symbols \((2pq/l)=-(q/l)=1\). However, it appears that to establish that such a prime congruent to \(1\mod 4\) exists, the Chinese Remainder Theorem must be used.NEWLINENEWLINE 2) In the abstract, the authors claim that each strongly ambiguous ideal class of \(k/\mathbb{Q}(i)\) capitulates in \(k^{\ast}\). However, aside from the problems with the proof of Theorem 5.6 mentioned above, it appears that this claim requires the assumptions of Theorem 6.2, which are that \(p\equiv 1\mod{8}\), \(q\equiv 3\mod{8}\), and \((p/q)=-1\), which implies that the \(2\)-class group of \(k\) is of type \((2,2,2)\).NEWLINENEWLINE 3) There are some minor notational errors in the paper, in the proofs of Proposition 3.3, Theorem 5.3, and Lemma 6.1. It should be stated in the abstract that \(F=\mathbb{Q}(i)\), and clarified that \(k^{\ast}\) is contained in (as opposed to ``is smaller than'') the relative genus field \((k/\mathbb{Q}(i))^{\ast}\).