Galois group

Not to be confused with Galois fields.

In Galois theory, a branch of abstract algebra, the Galois group of a certain type of field extension is a symmetry group characterizing how it extends the base field. Each element of the Galois group is a transformation of the field extension which leaves each element of the base field fixed.

This connection between fields and groups, given by the fundamental theorem of Galois theory, allows for group-theoretic tools to be used on problems in field theory, such as describing the solutions to quintic polynomials. The study of field extensions and their relationship to the polynomials that give rise to them via Galois groups is called Galois theory, so named in honor of Évariste Galois who first discovered them.

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