View source for Normal extension

''of a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067500/n0675001.png" />''

An algebraic field extension (cf. [[Extension of a field|Extension of a field]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067500/n0675002.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067500/n0675003.png" /> satisfying one of the following equivalent conditions:

1) any imbedding of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067500/n0675004.png" /> in the [[Algebraic closure|algebraic closure]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067500/n0675005.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067500/n0675006.png" /> comes from an automorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067500/n0675007.png" />;

2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067500/n0675008.png" /> is the splitting field of some family of polynomials with coefficients in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067500/n0675009.png" /> (cf. [[Splitting field of a polynomial|Splitting field of a polynomial]]);

3) any polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067500/n06750010.png" /> with coefficients in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067500/n06750011.png" />, irreducible over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067500/n06750012.png" /> and having a root in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067500/n06750013.png" />, splits in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067500/n06750014.png" /> into linear factors.

For every algebraic extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067500/n06750015.png" /> there is a maximal intermediate subfield <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067500/n06750016.png" /> that is normal over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067500/n06750017.png" />; this is the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067500/n06750018.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067500/n06750019.png" /> ranges over all imbeddings of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067500/n06750020.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067500/n06750021.png" />. There is also a unique minimal normal extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067500/n06750022.png" /> containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067500/n06750023.png" />. This is the composite of all fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067500/n06750024.png" />. It is called the normal closure of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067500/n06750025.png" /> relative to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067500/n06750026.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067500/n06750027.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067500/n06750028.png" /> are normal extensions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067500/n06750029.png" />, then so are the intersection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067500/n06750030.png" /> and the composite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067500/n06750031.png" />. However, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067500/n06750032.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067500/n06750033.png" /> are normal extensions, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067500/n06750034.png" /> need not be normal.

For fields of characteristic zero every normal extension is a Galois extension. In general, a normal extension is a [[Galois extension|Galois extension]] if and only if it is separable (cf. [[Separable extension|Separable extension]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  B.L. van der Waerden,   "Algebra" , '''1–2''' , Springer  (1967–1971)  (Translated from German)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S. Lang,   "Algebra" , Addison-Wesley  (1984)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M.M. Postnikov,   "Fundamentals of Galois theory" , Noordhoff  (1962)  (Translated from Russian)</TD></TR></table>