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 in L.[1][2] These are one of the conditions for algebraic extensions to be a Galois extension. Bourbaki calls such an extension a quasi-Galois extension.


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:[3]

Other properties

Let L be an extension of a field K. Then:

Equivalent conditions for normality

Let be algebraic. The field L is a normal extension if and only if any of the equivalent conditions below hold.

Examples and counterexamples

For example, 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 Let be a primitive cubic root of unity. Then since,

the map
is an embedding of in whose restriction to is the identity. However, is not an automorphism of

For any prime the extension is normal of degree It is a splitting field of Here denotes any th primitive root of unity. The field is the normal closure (see below) of

Normal closure

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.

See also


  1. ^ Lang 2002, p. 237, Theorem 3.3, NOR 3.
  2. ^ Jacobson 1989, p. 489, Section 8.7.
  3. ^ Lang 2002, p. 237, Theorem 3.3.
  4. ^ a b Lang 2002, p. 238, Theorem 3.4.