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Contents

   



(Top)
 


1 Definition  





2 Other properties  





3 Equivalent conditions for normality  





4 Examples and counterexamples  





5 Normal closure  





6 See also  





7 Citations  





8 References  














Normal extension






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From Wikipedia, the free encyclopedia
 


Inabstract algebra, a normal extension is an algebraic field extension L/K for which every irreducible polynomial over K that has a rootinL splits into linear factors in L.[1][2] 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.

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 closureofK). Then the following conditions, any of which can be regarded as a definition of normal extension, are equivalent:[3]

Other properties[edit]

Let L be an extension of a field K. Then:

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.

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 ofalgebraic numbers is the algebraic closure of and thus it contains Let be a primitive cubic root of unity. Then since,

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 thprimitive root of unity. The field is the normal closure (see below) of

Normal closure[edit]

IfK is a field and L is an algebraic extension of K, then there is some algebraic extension MofL 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 KisM itself. This extension is called the normal closure of the extension LofK.

IfL is a finite extensionofK, then its normal closure is also a finite extension.

See also[edit]

Citations[edit]

  1. ^ Lang 2002, p. 237, Theorem 3.3, NOR 3.
  • ^ Jacobson 1989, p. 489, Section 8.7.
  • ^ Lang 2002, p. 237, Theorem 3.3.
  • ^ a b Lang 2002, p. 238, Theorem 3.4.
  • References[edit]


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    This page was last edited on 2 May 2024, at 14:34 (UTC).

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