Galois Extensions

  • Joseph Rotman
Part of the Universitext book series (UTX)


The discussion of Galois groups began with a pair of fields, namely, an extension E/F that is a splitting field of some polynomial f(x) ∈ F[x]. Suppose that G = Gal(E/F); it is easy to see that
$$ F \subset {E^G} \subset E $$
. A natural question is whether F = E G ; in general, the answer is no. For example, if F = ℚ and E = (α), where α is the real cube root of 2, then G = Gal(E/F) = Gal( (α)/ℚ) = {1} (if σ ∈ G, then σ(α) is a root of x3 - 2; but E does not contain the other two (complex) roots of this polynomial). Hence E G = EF.


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Copyright information

© Springer-Verlag New York Inc. 1990

Authors and Affiliations

  • Joseph Rotman
    • 1
  1. 1.Department of MathematicsUniversity of Illinois at Urbana-ChampaignUrbanaUSA

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