Rationality questions: examples

  • Jean-Pierre Serre
Part of the Graduate Texts in Mathematics book series (GTM, volume 42)


Let G be a finite group of order g, and let m be a multiple of the orders of all the elements of G. Take as ground field K the field Q of rational numbers, and let Q(m) be the field obtained by adjoining the mth roots of unity to Q. The Galois group of Q(m) over Q is the group denoted Γ Q in 12.4; it is a subgroup of the group (Z/mZ)*. In fact:

Theorem (Gauss). We have Γ Q = (Z/mZ)*.


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Copyright information

© Springer-Verlag, New York Inc. 1977

Authors and Affiliations

  • Jean-Pierre Serre
    • 1
  1. 1.Chaire d’algèbre et géométrieCollège de FranceParisFrance

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