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The Galois group of a separable polynomial
Definition
Let be separable, and let be a splitting field of (Polynomials that split and splitting fields of a polynomial or a family of polynomials). By Equivalent characterizations of a finite Galois extension, the extension is finite Galois. The Galois group of over is
An ordering of the roots identifies with a permutation group. A different ordering conjugates that subgroup in the corresponding symmetric group. Isomorphisms between splitting fields exist by A base-field isomorphism extends to an isomorphism between splitting fields of corresponding polynomials and conjugate their automorphism groups. Thus the abstract group, and its root action up to relabelling, do not depend on the chosen splitting field.
Depends on
- Polynomials that split and splitting fields of a polynomial or a family of polynomials
- Finite Galois extensions and $\operatorname{Gal}(K/F)$
- Equivalent characterizations of a finite Galois extension
- A base-field isomorphism extends to an isomorphism between splitting fields of corresponding polynomials
Used by
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Sources
- J. S. Milne, Fields and Galois Theory, v5.10, The Galois group of a polynomial (standard reference, not scraped)
- K. Conrad, Galois Groups of Cubics and Quartics, Section 1 (standard reference, not scraped)