Galois group

In mathematics, a Galois group is a group associated with a certain type of field extension.

Suppose G is an extension of the field F, and consider the set of all field automorphisms of G which fix F pointwise. This set of automorphisms forms a group H. If there are no elements of G \ F which are fixed by all members of H, then the extension G/F is called a Galois extension, and H is the Galois group of the extension and is usually denoted Gal(G/F). It can be shown that G is algebraic over F if and only if the Galois group is pro-finite.

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We need a link to Galois theory and the fundamental theorem here.

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/Talk?