Template:Short description {{#invoke:other uses|otheruses}} 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.Template:SfnTemplate:Sfn 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.
DefinitionEdit
Let <math>L/K</math> be an algebraic extension (i.e., L is an algebraic extension of K), such that <math>L\subseteq \overline{K}</math> (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:Template:Sfn
- Every embedding of L in <math>\overline{K}</math> over K induces an automorphism of L.
- L is the splitting field of a family of polynomials in <math>K[X]</math>.
- Every irreducible polynomial of <math>K[X]</math> that has a root in L splits into linear factors in L.
Other propertiesEdit
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.Template:Sfn
- 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.Template:Sfn
Equivalent conditions for normalityEdit
Let <math>L/K</math> 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 <math>S \subseteq K[x]</math> of polynomials that each splits over L, such that if <math>K\subseteq F\subsetneq L</math> are fields, then S has a polynomial that does not split in F;
- All homomorphisms <math>L \to \bar{K}</math> that fix all elements of K have the same image;
- The group of automorphisms, <math>\text{Aut}(L/K),</math> of L that fix all elements of K, acts transitively on the set of homomorphisms <math>L \to \bar{K}</math> that fix all elements of K.
Examples and counterexamplesEdit
For example, <math>\Q(\sqrt{2})</math> is a normal extension of <math>\Q,</math> since it is a splitting field of <math>x^2-2.</math> On the other hand, <math>\Q(\sqrt[3]{2})</math> is not a normal extension of <math>\Q</math> since the irreducible polynomial <math>x^3-2</math> has one root in it (namely, <math>\sqrt[3]{2}</math>), but not all of them (it does not have the non-real cubic roots of 2). Recall that the field <math>\overline{\Q}</math> of algebraic numbers is the algebraic closure of <math>\Q,</math> and thus it contains <math>\Q(\sqrt[3]{2}).</math> Let <math>\omega</math> be a primitive cubic root of unity. Then since, <math display="block">\Q (\sqrt[3]{2})=\left. \left \{a+b\sqrt[3]{2}+c\sqrt[3]{4}\in\overline{\Q }\,\,\right | \,\,a,b,c\in\Q \right \}</math> the map <math display="block">\begin{cases} \sigma:\Q (\sqrt[3]{2})\longrightarrow\overline{\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}</math> is an embedding of <math>\Q(\sqrt[3]{2})</math> in <math>\overline{\Q}</math> whose restriction to <math>\Q </math> is the identity. However, <math>\sigma</math> is not an automorphism of <math>\Q (\sqrt[3]{2}).</math>
For any prime <math>p,</math> the extension <math>\Q (\sqrt[p]{2}, \zeta_p)</math> is normal of degree <math>p(p-1).</math> It is a splitting field of <math>x^p - 2.</math> Here <math>\zeta_p</math> denotes any <math>p</math>th primitive root of unity. The field <math>\Q (\sqrt[3]{2}, \zeta_3)</math> is the normal closure (see below) of <math>\Q (\sqrt[3]{2}).</math>
Normal closureEdit
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.