×

Central extensions and rational quadratic forms. (English) Zbl 0771.11040

It is characterized by means of simple quadratic forms the set of rational primes that are decomposed completely in a non-abelian central extension which is abelian over a quadratic field. More precisely, let \(L=\mathbb{Q}(\sqrt{d_ 1},\sqrt{d_ 2})\) be a bicyclic biquadratic field, and let \(K=\mathbb{Q}(\sqrt{d_ 1d_ 2})\). Denote by \(S_ K(\widetilde m)\) the ray class field mod\( m\) of \(K\) in narrow sense for a large rational integer \(m\). Let \(L_ m^*\) be the maximal abelian extension over \(\mathbb{Q}\) contained in \(S_ K(\widetilde m)\) and \(\widehat L_ m\) be the maximal extension contained in \(S_ K(\widetilde m)\) such that \(\text{Gal}(\widehat L_ m/L)\) is contained in the center of \(\text{Gal}(\widehat L_ m/\mathbb{Q})\). Then it is proved that any rational prime \(p\) not dividing \(d_ 1 d_ 2 m\) is decomposed completely in \(L_ m^*/\mathbb{Q}\) if and only if \(p\) is representable by rational integers \(x\) and \(y\) such that \(x\equiv 1\) and \(y\equiv 0\bmod m\) as follows \[ p={{ax^ 2+bxy+cy^ 2} \over a}, \] where \(a\), \(b\), \(c\) are rational integers such that \(b^ 2-4ac\) is equal to the discriminant of \(K\) and (a) is a norm of a representative of the ray class group of \(K\bmod m\).
Moreover \(p\) is decomposed completely in \(\widehat L_ m/L_ m^*\) if and only if \(({{d_ 1}\over a})=1\).

MSC:

11R20 Other abelian and metabelian extensions
11R44 Distribution of prime ideals
11E12 Quadratic forms over global rings and fields
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Nagoya Math. J. 93 pp 61– (1984) · Zbl 0526.12009 · doi:10.1017/S0027763000020730
[2] Nagoya Math. J. 98 pp 77– (1985) · Zbl 0562.12012 · doi:10.1017/S0027763000021371
[3] Nagoya Math. J. 66 pp 167– (1977) · Zbl 0337.12011 · doi:10.1017/S0027763000017797
[4] Nagoya Math. J. 79 pp 79– (1980) · Zbl 0444.12001 · doi:10.1017/S0027763000018948
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.