Condition nécessaire et suffisante pour que certain groupe de Galois soit métacyclique

Azizi, Abdelmalek; Taous, Mohammed

Annales mathématiques Blaise Pascal, Tome 16 (2009) no. 1, pp. 83-92.

Soient $d$ est un entier sans facteurs carrés, $\mathbf{K}=\mathbf{Q}(\sqrt{d},i)$, $i=\sqrt{-1}$, ${\mathbf{K}}_2^{\left(1\right)}$ le $2$-corps de classes de Hilbert de $\mathbf{K}$, ${\mathbf{K}}_2^{\left(2\right)}$ le $2$-corps de classes de Hilbert de ${\mathbf{K}}_2^{\left(1\right)}$ et $G=\mathrm{Gal}({\mathbf{K}}_2^{\left(2\right)}/\mathbf{K})$ le groupe de Galois de ${\mathbf{K}}_2^{\left(2\right)}/\mathbf{K}$. Notre but est de montrer qu’il existe une forme de $d$ tel que le $2$-groupe $G$ est non métacyclique et de donner une condition nécessaire et suffisante pour que le groupe $G$ soit métacyclique dans le cas où $d=2p$ avec $p$ un nombre premier tel que $p\equiv 1(mod4)$.

Let $d$ be positive square-free integers, $\mathbf{K}=\mathbf{Q}(\sqrt{d},i)$ and $i=\sqrt{-1}$. Let ${\mathbf{K}}_1^{\left(2\right)}$ be the Hilbert $2$-class field of $\mathbf{K}$, ${\mathbf{K}}_2^{\left(2\right)}$ be the Hilbert $2$-class field of ${\mathbf{K}}_1^{\left(2\right)}$ and $G=\mathrm{Gal}({\mathbf{K}}_2^{\left(2\right)}/\mathbf{K})$ be the Galois group of ${\mathbf{K}}_2^{\left(2\right)}/\mathbf{K}$. Our goal is to show that there is some form of $d$ such $G$ is a nonmetacyclic $2$-group and give the necessary condition and sufficient for the group $G$ to be metacyclic in case $d=2p$ with $p$ a prime number such that $p\equiv 1(mod4)$.

[1] Azizi, A. Capitulation of the $2$-ideal Classes of $\mathbb{Q}(\sqrt{p_1{p}_2},i)$ Where $p_1$ and $p_2$ are primes such that $p_1\equiv 1(mod8)$, $p_2\equiv 5(mod8)$ and $\left(\frac{p_1}{p_2}\right)=-1$, Lecture notes in pure and applied mathematics, Volume 208 (1999), pp. 13-19 | MR | Zbl

[2] Azizi, A. Sur une question de Capitulation, Proc. Amer. Math. Soc, Volume 130 (2002), pp. 2197-2202 | DOI | MR | Zbl

[3] Azizi, A.; Taous, M. Capitulation of $2$-ideal classes of $\mathbf{k}=\mathbb{Q}(\sqrt{2p},i)$ in the genus field of $\mathbf{k}$ where $p$ is prime such that $p\equiv 1(mod8)$, IJPAM, Volume 35 (2007) no. 2, pp. 481-487 | MR

[4] Baginski, C.; Konovalov, A. On $2$-groups of almost maximal class, Publ. Math, Volume 65 (2004) no. 1-2, pp. 97-131 | MR | Zbl

[5] Benjamin, E.; Lemmermeyer, F.; Snyder, C. Imaginary Quadratic Fields $k$ with Cyclic $C{l}_2\left(k^1\right)$, J. Number Theory, Volume 67 (1997), pp. 229-245 | DOI | MR | Zbl

[6] Benjamin, E.; Lemmermeyer, F.; Snyder, C. Real quadratic fields with abelian $2$-class field tower, J. Number Theory, Volume 73 (1998), pp. 182-194 | DOI | MR | Zbl

[7] Benjamin, E.; Lemmermeyer, F.; Snyder, C. Imaginary quadratic fields with $C{l}_2\left(k\right)=(2,2^m)$ and rank $C{l}_2\left(k^1\right)=2$, Pac. J. Math, Volume 198 (2001), pp. 15-31 | DOI | MR | Zbl

[8] Benjamin, E.; Snyder, C. Number Fields with $2$-class Number Isomorphic to $(2,2^m)$ (1994) (preprint)

[9] Blackburn, N. On Prime Power Groups in which the Derived Group has Two Generators, Proc. Cambridge Phil. Soc, Volume 53 (1957), pp. 19-27 | DOI | MR | Zbl

[10] Blackburn, N. On a special class of $p$-groups, Acta Math, Volume 100 (1958), pp. 45-92 | DOI | MR | Zbl

[11] James, R. $2$-Groups of Almost Maximal Class, J. Austral. Math. Soc. Ser. A, Volume 19 (1975), pp. 343-357 | DOI | MR | Zbl

[12] Kisilevsky, H. Number fields with class number congruent to $4$ $\mathrm{mod}$ $8$ and Hilbert’s theorem $94$, J. Number Theory, Volume 8 (1976) no. 3, pp. 271-279 | DOI | MR | Zbl

[13] Kubota, T. Über die Beziehung der Klassenzahlen der Unterkörper des bizyklischen Zahlkörpers, Nagoya Math. J, Volume 6 (1953), pp. 119-127 | MR | Zbl

[14] McCall, T.M.; Parry, C.J.; Ranalli, R.R. On imaginary bicyclic biquadratic fields with cyclic $2$-class group, J. Number Theory, Volume 53 (1995), pp. 88-99 | DOI | MR | Zbl

[15] Taussky, O. A Remark on the Class Field Tower, J. Number Theory, Volume 12 (1937), pp. 82-85