In abstract algebra, a normal extension is an algebraic field extension L/K for which every irreducible polynomial over K that has a root in L splits into linear factors over L.[1][2] This is one of the conditions for an algebraic extension to be a Galois extension.
Bourbaki calls such an extension a quasi-Galois extension.
For finite extensions, a normal extension is identical to a splitting field.
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:[3] Let L be an extension of a field K. Then: Let
The field L is a normal extension if and only if any of the equivalent conditions below hold.
is a normal extension of
since it is a splitting field of
On the other hand,
is not a normal extension of
since the irreducible polynomial
has one root in it (namely,
), but not all of them (it does not have the non-real cubic roots of 2).
Recall that the field
of algebraic numbers is the algebraic closure of
and thus it contains
ω
be a primitive cubic root of unity.
σ :
⟼ a + b ω
ω
is an embedding of
σ
is normal of degree
It is a splitting field of
th primitive root of unity.
is the normal closure (see below) of
If K is a field and L is an algebraic extension of K, then there is some algebraic extension M of L such that M is a normal extension of K. Furthermore, up to isomorphism there is only one such extension that is minimal, that is, the only subfield of M that contains L and that is a normal extension of K is M itself.
This extension is called the normal closure of the extension L of K. If L is a finite extension of K, then its normal closure is also a finite extension.