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Splitting Fields

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

Abstract

We consider field extensions which are generated by all zeros of a given polynomial over a field containing its coefficients. Such a field is called a splitting field. Splitting fields of polynomials play a central role in Galois theory. It is intuitively clear that a splitting field of a polynomial is a very natural object carrying a lot of information about it. We will find a confirmation of this in the following chapters, in particular, when we come to the main theorems of Galois theory (Chap.  9) and its applications in subsequent chapters (e.g. in Chap.  13, where we study the solvability of equations by radicals). As an important example, we study finite fields in this chapter. These can be easily described as splitting fields of very simple polynomials over finite prime fields. We further consider the notion of an algebraic closure of a field K, which is a minimal field extension of K, containing a splitting field of every polynomial with coefficients in K.

References

  1. [Sch]
    A. Schinzel, Polynomials with Special Regard to Reducibility, Encyclopedia of Mathematics and its Applications 77, Cambridge University Press, 2000.Google Scholar
  2. [Tsch]
    N.G. Tschebotaröw, Grundzüge der Galois’schen Theorie, H. Schwerdtfeger (ed.), P. Noordhoff, 1950.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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