Abstract
Let \(d\) be a square-free integer, \(\mathbf {k}=\mathbb {Q}(\sqrt{d},\,i)\) and \(i=\sqrt{-1}\). Let \(\mathbf {k}_1^{(2)}\) be the Hilbert \(2\)-class field of \(\mathbf {k}\), \(\mathbf {k}_2^{(2)}\) be the Hilbert \(2\)-class field of \(\mathbf {k}_1^{(2)}\) and \(G=\mathrm {Gal}(\mathbf {k}_2^{(2)}/\mathbf {k})\) be the Galois group of \(\mathbf {k}_2^{(2)}/\mathbf {k}\). We give necessary and sufficient conditions to have \(G\) metacyclic in the case where \(d=pq\), with \(p\) and \(q\) primes such that \(p\equiv 1\pmod 8\) and \(q\equiv 5\pmod 8\) or \(p\equiv 1\pmod 8\) and \(q\equiv 3\pmod 4\).
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References
A. Azizi, Unités de certains corps de nombres imaginaires et abéliens sur \({\mathbb{Q}}\). Ann. Sci. Math. Québec 23, 87–93 (1999)
A. Azizi, Capitulation of the 2-ideal classes of \({\mathbb{Q}}(\sqrt{p_1p_2},\, i) \) where \(p_1\) and \(p_2\) are primes such thet \(p_1\equiv 1 \text{ mod } 8\), \(p_2\equiv 5 \text{ mod } 8\) and \((\frac{p_1}{p_2})=-1\). Lect. Notes Pure Appl. Math. 208, 13–19 (1999)
A. Azizi, Sur le \(2\)-groupe de classe d’idéaux de \({\mathbb{Q}(\sqrt{d}, i)}\). Rend. Circ. Mat. Palmero 48(2), 71–92 (1999)
A. Azizi, Sur la capitulation des \(2\)-classes d’idéaux de \(k ={\mathbb{Q}}(\sqrt{2pq}, i)\), où \(p\equiv -q\equiv 1 \text{ mod } 4\). Acta. Arith. 94, 383–399 (2000)
A. Azizi, Sur une question de capitulation. Proc. Amer. Math. Soc. 130, 2197–2202 (2002)
A. Azizi, M. Taous, Determination des corps \(\mathbf{k}={\mathbb{Q}}(\sqrt{d}, i)\) dont le \(2\)-groupes de classes est de type \((2, 4)\) ou \((2, 2, 2)\). Rend. Istit. Mat. Univ. Trieste. 40, 93–116 (2008)
A. Azizi, M. Taous, Condition nécessaire et suffisante pour que certain groupe de Galois soit métacyclique. Ann. Math. Blaise Pascal. 16(1), 83–92 (2009)
A. Scholz, Über die Lösbarkeit der Gleichung \(t^2-Du^2=-4\). Math. Z. 39, 95–111 (1934)
C. Chevalley, Sur la théorie du corps de classes dans les corps finis et les corps locaux. J. Fac. Sci. Tokyo Sect. 1 t. 2, 365–476 (1933)
E. Benjamin, F. Lemmermeyer, C. Snyder, Real quadratic fields with abelian \(2\)-class field tower. J. Number Theory. 73, 182–194 (1998)
F. Lemmermeyer, Reciprocity laws, springer monographs in mathematics (Springer-Verlag, Berlin, 2000)
G. Gras, Class field theory, from theory to practice (Springer Verlag, New York, 2003)
H. Kisilevsky, Number fields with class number ongruent to \(4 \text{ mod } 8\) and Hilbert’s thorem \(94\). J. Number Theory. 8, 271–279 (1976)
H. Kurzweil, B. Stellmacher, The theory of finite groups: an introduction (Springer-Verlag, New York, 2004)
K. Miyake, Algebraic investigations oh Hilbert’s theorem \(94\), the principal ideal theorem and capitulation problem. Expos. Math. 7, 289–346 (1989)
T.M. McCall, C.J. Parry, R.R. Ranalli, On imaginary bicyclic biquadratic fields with cyclic \(2\)-class group. J. Number Theory. 53, 88–99 (1995)
Y. Berkovich, Z. Janko, On subgroups of finite \(p\)-groups. Israel J. Math. 171, 29–49 (2009)
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Azizi, A., Taous, M. & Zekhnini, A. On the rank of the \(2\)-class group of \(\mathbb {Q}(\sqrt{p}, \sqrt{q},\sqrt{-1} )\) . Period Math Hung 69, 231–238 (2014). https://doi.org/10.1007/s10998-014-0049-9
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DOI: https://doi.org/10.1007/s10998-014-0049-9