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 Ka be an algebraic closure of K containing L.
Every embedding σ of L in Ka 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.[citation needed]
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.[citation needed]
Examples and counterexamples
For example, \( \mathbb{Q} ({\sqrt {2}}) \) is a normal extension of \( {\displaystyle \mathbb {Q} ,} \) since it is a splitting field of \( {\displaystyle x^{2}-2.} \) On the other hand, \( {\displaystyle \mathbb {Q} ({\sqrt[{3}]{2}})} \) is not a normal extension of \( \mathbb {Q} \) since the irreducible polynomial \( {\displaystyle x^{3}-2} \) has one root in it (namely, \( {\sqrt[{3}]{2}} \) ), but not all of them (it does not have the non-real cubic roots of 2). Recall that the field \( {\displaystyle {\overline {\mathbb {Q} }}} \) of algebraic numbers is the algebraic closure of \( {\displaystyle \mathbb {Q} ,} \) i.e., it contains \( {\displaystyle \mathbb {Q} ({\sqrt[{3}]{2}}).} \) Since,
\( {\displaystyle \mathbb {Q} ({\sqrt[{3}]{2}})=\left.\left\{a+b{\sqrt[{3}]{2}}+c{\sqrt[{3}]{4}}\in {\overline {\mathbb {Q} }}\,\,\right|\,\,a,b,c\in \mathbb {Q} \right\}} \)
and, if ω is a primitive cubic root of unity, then the map
\( {\displaystyle {\begin{cases}\sigma :\mathbb {Q} ({\sqrt[{3}]{2}})\longrightarrow {\overline {\mathbb {Q} }}\\a+b{\sqrt[{3}]{2}}+c{\sqrt[{3}]{4}}\longmapsto a+b\omega {\sqrt[{3}]{2}}+c\omega ^{2}{\sqrt[{3}]{4}}\end{cases}}} \)
is an embedding of \( {\displaystyle \mathbb {Q} ({\sqrt[{3}]{2}})} \) in \( {\displaystyle {\overline {\mathbb {Q} }}} \) whose restriction to
\( \mathbb{Q} \) is the identity. However, σ is not an automorphism of \( {\displaystyle \mathbb {Q} ({\sqrt[{3}]{2}})}. \)
For any prime p, the extension \( {\displaystyle \mathbb {Q} ({\sqrt[{p}]{2}},\zeta _{p})} \) is normal of degree p(p − 1). It is a splitting field of xp − 2. Here \( \zeta _{p} \) denotes any pth primitive root of unity. The field \( {\displaystyle \mathbb {Q} ({\sqrt[{3}]{2}},\zeta _{3})} \) is the normal closure (see below) of \( {\displaystyle \mathbb {Q} ({\sqrt[{3}]{2}})}. \)
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
Galois extension
Normal basis
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
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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