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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]} 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.

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]}

- Every embedding of
*L*in over*K*induces an automorphism of*L*. *L*is the splitting field of a family of polynomials in .- Every irreducible polynomial of that has a root in
*L*splits into linear factors in*L*.

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

- If
*L*is a normal extension of*K*and if*E*is an intermediate extension (that is,*L*⊇*E*⊇*K*), then*L*is a normal extension of*E*.^{[4]} - If
*E*and*F*are normal extensions of*K*contained in*L*, then the compositum*EF*and*E*∩*F*are also normal extensions of*K*.^{[4]}

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

- The minimal polynomial over
*K*of every element in*L*splits in*L*; - There is a set of polynomials that each splits over
*L*, such that if are fields, then*S*has a polynomial that does not split in*F*; - All homomorphisms that fix all elements of
*K*have the same image; - The group of automorphisms, of
*L*that fix all elements of*K*, acts transitively on the set of homomorphisms that fix all elements of*K*.

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

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.

- Lang, Serge (2002),
*Algebra*, Graduate Texts in Mathematics, vol. 211 (Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4, MR 1878556 - Jacobson, Nathan (1989),
*Basic Algebra II*(2nd ed.), W. H. Freeman, ISBN 0-7167-1933-9, MR 1009787