Cyclotomic Extensions

  • Juliusz Brzeziński
Chapter
Part of the Springer Undergraduate Mathematics Series book series (SUMS)

Abstract

This chapter illustrates the general theory of Galois extensions in a special case. We study field extensions, mostly of the rational numbers, generated by the roots of 1. Even if such fields are simple to describe in purely algebraic terms, they are rich as mathematical objects. We explore some of their properties, which find different applications in number theory and algebra. Among many applications, there is a proof of a special case of Dirichlet’s theorem on primes in arithmetic progression using the cyclotomic polynomials. This chapter also includes an exercise on a solution of the inverse Galois problem for all abelian groups.

References

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    D. Gay, W.Y. Vélez, On the degree of the splitting field of an irreducible binomial, Pacific J. Math. 78 (1978), 117–120.MathSciNetCrossRefMATHGoogle Scholar
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    H. Koch, Number Theory, Algebraic Numbers and Functions, Graduate Studies in Mathematics vol. 24, AMS, 2000.Google Scholar
  3. [S2]
    J.-P. Serre, A Course in Arithmetic, Graduate Texts in Mathematics vol. 7, Springer, 2006.Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Juliusz Brzeziński
    • 1
    • 2
  1. 1.Department of Mathematical SciencesUniversity of GothenburgGöteborgSweden
  2. 2.Chalmers University of TechnologyGöteborgSweden

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