Abstract
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
. A natural question is whether F = EG; in general, the answer is no. For example, if F = ℚ and E = ℚ (α), where α is the real cube root of 2, then G = Gal(E/F) = Gal(ℚ (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 EG = E ≠F.
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© 1990 Springer-Verlag New York Inc.
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Rotman, J. (1990). Galois Extensions. In: Galois Theory. Universitext. Springer, New York, NY. https://doi.org/10.1007/978-1-4684-0367-1_15
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DOI: https://doi.org/10.1007/978-1-4684-0367-1_15
Publisher Name: Springer, New York, NY
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