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Concept# Normal extension

Summary

In abstract algebra, a normal extension is an algebraic field extension L/K for which every irreducible polynomial over K which has a root in L, splits into linear factors in L. 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:
Every embedding of L in induces an automorphism of L.
L is the splitting field of a family of polynomials in .
Every irreducible polynomial of which 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.
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.
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 simultaneously split over L, such that if are fields, then S has a polynomial that does not split in F;
All homomorphisms have the same image;
The group of automorphisms, of L which fixes elements of K, acts transitively on the set of homomorphisms
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 that is, it contains Since,
and, if is a primitive cubic root of unity, then the map
is an embedding of in whose restriction to is the identity.

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Related concepts (8)

Galois extension

In mathematics, a Galois extension is an algebraic field extension E/F that is normal and separable; or equivalently, E/F is algebraic, and the field fixed by the automorphism group Aut(E/F) is precisely the base field F. The significance of being a Galois extension is that the extension has a Galois group and obeys the fundamental theorem of Galois theory. A result of Emil Artin allows one to construct Galois extensions as follows: If E is a given field, and G is a finite group of automorphisms of E with fixed field F, then E/F is a Galois extension.

Normal extension

In abstract algebra, a normal extension is an algebraic field extension L/K for which every irreducible polynomial over K which has a root in L, splits into linear factors in L. 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).

Separable extension

In field theory, a branch of algebra, an algebraic field extension is called a separable extension if for every , the minimal polynomial of over F is a separable polynomial (i.e., its formal derivative is not the zero polynomial, or equivalently it has no repeated roots in any extension field). There is also a more general definition that applies when E is not necessarily algebraic over F. An extension that is not separable is said to be inseparable.

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