Galois group of a polynomial: Difference between revisions

Revision as of 00:45, 8 November 2023

This article defines a natural context where a group occurs, or is associated, with another algebraic, topological, analytic or discrete structure
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This term is related to: Galois theory
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Definition

For $K$ a field, the Galois group of a polynomial $f(X)\in K[X]$ is defined to be the Galois group (in the sense of field extensions) of a splitting field for $f(X)$ over $K$ .

Examples

The polynomial $f(X)=(X^{2}-2)(X^{2}-3)$ has splitting field $\mathbb {Q} ({\sqrt {2}},{\sqrt {3}})$ over $\mathbb {Q}$ . It can be shown that this field extension has Galois group isomorphic to the Klein four-group.

The polynomial $X^{n}-X-1$ has Galois group the symmetric group on $n$ letters.

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 * The polynomial <math>f(X) = (X^2-2)(X^2-3)</math> has splitting field <math>\Q(\sqrt{2}, \sqrt{3})</math> over <math>\Q</math>. It can be shown that this field extension has Galois group isomorphic to the [[Klein four-group]].
-* The polynomial <math>X^n-X-1</math> is the [[symmetric group]] on <math>n</math> letters.
+* The polynomial <math>X^n-X-1</math> has Galois group the [[symmetric group]] on <math>n</math> letters.
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