In
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The ter ...
, a normal extension is an
algebraic field extension ''L''/''K'' for which every
irreducible polynomial over ''K'' which has a root in ''L'', splits into linear factors in ''L''. These are one of the conditions for algebraic extensions to be a
Galois extension
In mathematics, a Galois extension is an algebraic field extension ''E''/''F'' that is normal and separable; or equivalently, ''E''/''F'' is algebraic, and the field fixed by the automorphism group Aut(''E''/''F'') is precisely the base field ' ...
.
Bourbaki calls such an extension a quasi-Galois extension.
Definition
Let ''
'' be an algebraic extension (i.e. ''L'' is an algebraic extension of ''K''), such that
(i.e. ''L'' is contained in an
algebraic closure of ''K''). Then the following conditions, any of which can be regarded as a definition of ''normal extension'', are equivalent:
* Every
embedding
In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup.
When some object X is said to be embedded in another object Y, the embedding is g ...
of ''L'' in
induces an automorphism of ''L''.
* ''L'' is the splitting field of a family of polynomials in