Normal extension

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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.[1][2] These are one of the conditions for algebraic extensions to be a Galois extension. Bourbaki calls such an extension a quasi-Galois extension.

Definition[edit]

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, which any of them can be regarded as a definition of normal extension, are equivalent:[3]

  • 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.

Other properties[edit]

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]

Equivalent conditions for normality[edit]

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

Examples and counterexamples[edit]

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. 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

Normal closure[edit]

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, that is, 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[edit]

Citations[edit]

  1. ^ Lang 2002, p. 237, Theorem 3.3, NOR 3.
  2. ^ Jacobson 1989, p. 489, Section 8.7.
  3. ^ Lang 2002, p. 237, Theorem 3.3.
  4. ^ a b Lang 2002, p. 238, Theorem 3.4.

References[edit]

  • 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