# Normal extension

In abstract algebra, a **normal extension** is an algebraic field extension *L*/*K* for which every polynomial that is irreducible over *K* either has no root in *L* or splits into linear factors in *L*. Bourbaki calls such an extension a **quasi-Galois extension**.

## Definition

The algebraic field extension *L*/*K* is normal (we also say that *L* is normal over *K*) if every irreducible polynomial over K that has at least one root in *L* splits over *L*. In other words, if *α* ∈ *L*, then all conjugates of *α* over *K* (i.e., all roots of the minimal polynomial of *α* over *K*) belong to *L*.

## Equivalent properties

The normality of *L*/*K* is equivalent to either of the following properties. Let *K*^{a} be an algebraic closure of *K* containing *L*.

- Every embedding
*σ*of*L*in*K*^{a}that restricts to the identity on*K*, satisfies σ(*L*) =*L*(*σ*is an automorphism of*L*over*K.*) - Every irreducible polynomial in
*K*[*X*] that has one root in*L*, has all of its roots in*L*, that is, it decomposes into linear factors in*L*[*X*]. (One says that the polynomial*splits*in*L*.)

If *L* is a finite extension of *K* that is separable (for example, this is automatically satisfied if *K* is finite or has characteristic zero) then the following property is also equivalent:

- There exists an irreducible polynomial whose roots, together with the elements of
*K*, generate*L*. (One says that*L*is the splitting field for the polynomial.)

## Other properties

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 (i.e.,*L*⊃*E*⊃*K*), then*L*is a normal extension of*E*. - 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*.

## 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 i.e., it contains Since,

and, if *ω* is a primitive cubic root of unity, then the map

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

For any prime *p*, the extension is normal of degree *p*(*p* − 1). It is a splitting field of *x ^{p}* − 2. Here denotes any

*p*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 which is minimal, i.e., the only subfield of *M* which contains *L* and which 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

## References

- Lang, Serge (2002),
*Algebra*, Graduate Texts in Mathematics,**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