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Galois extensions for symmetric group:S3

This article gives specific information, namely, Galois extensions, about a particular group, namely: symmetric group:S3.
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This article discusses Galois extensions of fields for which the Galois group is the symmetric group of degree three. Note that since the Galois group of any extension of finite fields is cyclic, there are no finite examples.

Contents

Adjoining all cube roots

Consider the field \mathbb{Q} of rational numbers. Let a \in \mathbb{Q} be an element that has no cube root in \mathbb{Q}. The splitting field of x^3 - a is an extension of degree six with Galois group equal to the symmetric group of degree three.

The structure of sub-extensions is as follows:

  • There is exactly one quadratic sub-extension: the extension obtained by adjoining a primitive cube root of unity, i.e., a root of the polynomial x^2 + x + 1. In the Galois correspondence, this corresponds to the unique subgroup of order three.
  • There are three cubic sub-extensions, each corresponding to adjoining a particular cube root.

The symmetric group of degree three acts naturally on the three cube roots, permuting them in any possible manner. One way of viewing this is as a subfield of \mathbb{C}, where one of the roots is real and the other two are complex conjugates. In this case, the restriction of complex conjugation to this field flips the two conjugates and fixes the real root. We can also do an automorphism that cyclically permutes the roots.