Osaka Journal of Mathematics

Construction of unramified extensions with a prescribed Galois group

Kwang-Seob Kim

Full-text: Open access


In this article, we shall prove that for any finite solvable group $G$, there exist infinitely many abelian extensions $K/\mathbb{Q}$ and Galois extensions $M/\mathbb{Q}$ such that the Galois group $\Gal(M/K)$ is isomorphic to $G$ and $M/K$ is unramified. The difference between our result and [3, 4, 6, 7, 13] is that we have a base field $K$ which is not only Galois over $\mathbb{Q}$, but also has very small degree compared to their results. We will also get another proof of Nomura's work [9], which gives us a base field of smaller degree than Nomura's. Finally for a given finite nonabelian simple group $G$, we will show there exists an unramified extension $M/K'$ such that the Galois group is isomorphic to $G$ and $K'$ has relatively small degree.

Article information

Osaka J. Math., Volume 52, Number 4 (2015), 1039-1051.

First available in Project Euclid: 18 November 2015

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 12F12: Inverse Galois theory
Secondary: 11R29: Class numbers, class groups, discriminants


Kim, Kwang-Seob. Construction of unramified extensions with a prescribed Galois group. Osaka J. Math. 52 (2015), no. 4, 1039--1051.

Export citation


  • B.N. Cooperstein: Minimal degree for a permutation representation of a classical group, Israel J. Math. 30 (1978), 213–235.
  • G. Cornell: Abhyankar's lemma and the class group; in Number Theory, Carbondale 1979 (Proc. Southern Illinois Conf., Southern Illinois Univ., Carbondale, Ill., 1979), Lecture Notes in Math. 751, Springer, Berlin, 1979, 82–88.
  • J. Elstrodt, F. Grunewald and J. Mennicke: On unramified $A_{m}$-extensions of quadratic number fields, Glasgow Math. J. 27 (1985), 31–37.
  • A. Fröhlich: On non-ramified extensions with prescribed Galois group, Mathematika 9 (1962), 133–134.
  • B. Huppert: Endliche Gruppen, I, Die Grundlehren der Mathematischen Wissenschaften 134, Springer, Berlin, 1967.
  • K.S. Kedlaya: A construction of polynomials with squarefree discriminants, Proc. Amer. Math. Soc. 140 (2012), 3025–3033.
  • T. Kondo: Algebraic number fields with the discriminant equal to that of a quadratic number field, J. Math. Soc. Japan 47 (1995), 31–36.
  • J. Neukirch, A. Schmidt and K. Wingberg: Cohomology of Number Fields, second edition, Grundlehren der Mathematischen Wissenschaften 323, Springer, Berlin, 2008.
  • A. Nomura: On the existence of unramified $p$-extensions with prescribed Galois group, Osaka J. Math. 47 (2010), 1159–1165.
  • M. Ozaki: Construction of maximal unramified $p$-extensions with prescribed Galois groups, Invent. Math. 183 (2011), 649–680.
  • K. Uchida: Unramified extensions of quadratic number fields, II, Tôhoku Math. J. (2) 22 (1970), 220–224.
  • L.C. Washington: Introduction to Cyclotomic Fields, Graduate Texts in Mathematics 83, Springer, New York, 1982.
  • Y. Yamamoto: On unramified Galois extensions of quadratic number fields, Osaka J. Math. 7 (1970), 57–76.