# Article

Full entry | PDF   (0.2 MB)
Keywords:
class group; class field tower; multiquadratic number field
Summary:
It is well known by results of Golod and Shafarevich that the Hilbert $2$-class field tower of any real quadratic number field, in which the discriminant is not a sum of two squares and divisible by eight primes, is infinite. The aim of this article is to extend this result to any real abelian $2$-extension over the field of rational numbers. So using genus theory, units of biquadratic number fields and norm residue symbol, we prove that for every real abelian $2$-extension over $\mathbb Q$ in which eight primes ramify and one of theses primes $\equiv -1\pmod 4$, the Hilbert $2$-class field tower is infinite.
References:
[1] III., F. Gerth: Some real quadratic fields with infinite Hilbert $2$-class field towers. Jap. J. Math., New Ser. 31 (2005), 175-181. DOI 10.4099/math1924.31.175 | MR 2153730 | Zbl 1075.11066
[2] Golod, E. S., Shafarevich, I. R.: On the class field tower. Izv. Akad. Nauk SSSR, Ser. Mat. 28 (1964), 261-272 Russian; English translation in Transl., Ser. 2, Am. Math. Soc. 48 (1965), 91-102. MR 0161852
[3] Hasse, H.: Neue Begründung und Verallgemeinerung der Theorie des Normenrestsymbols. J. f. M. 162 (1930), 134-144 German.
[4] Ishida, M.: The Genus Fields of Algebraic Number Fields. Lecture Notes in Mathematics 555. Springer Berlin (1976). MR 0435028
[5] Jehne, W.: On knots in algebraic number theory. J. Reine Angew. Math. 311-312 (1979), 215-254. MR 0549967 | Zbl 0432.12006
[6] Kuroda, S.: Über den Dirichletschen Körper. J. Fac. Sci. Univ. Tokyo, Sect. I 4 (1943), 383-406 German. MR 0021031 | Zbl 0061.05901
[7] Kuz'min, L. V.: Homologies of profinite groups, the Schur multiplicator and class field theory. Izv. Akad. Nauk. SSSR Ser. Mat. 33 (1969), 1220-1254 Russian. MR 0255511
[8] Maire, C.: A refinement of the Golod-Shafarevich theorem. (Un raffinement du théoreme de Golod-Šafarevič). Nagoya Math. J. 150 (1998), 1-11 French. MR 1633138
[9] Mouhib, A.: On the Hilbert $2$-class field tower of real quadratic fields. (Sur la tour des $2$-corps de classes de Hilbert des corps quadratiques réels). Ann. Sci. Math. Qu. 28 (2004), 179-187 French. MR 2183105

Partner of