Normal extension

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of a field $K$

An algebraic field extension (cf. Extension of a field) $L$ of $K$ satisfying one of the following equivalent conditions:

1) any imbedding of $L$ in the algebraic closure $\bar K$ of $K$ comes from an automorphism of $L$;

2) $L$ is the splitting field of some family of polynomials with coefficients in $K$ (cf. Splitting field of a polynomial);

3) any polynomial $f(x)$ with coefficients in $K$, irreducible over $K$ and having a root in $L$, splits in $L$ into linear factors.

For every algebraic extension $F/K$ there is a maximal intermediate subfield $L$ that is normal over $K$; this is the field $L = \bigcap_\sigma F^\sigma$, where $\sigma$ ranges over all imbeddings of $F$ in $\bar K$. There is also a unique minimal normal extension of $K$ containing $F$. This is the composite of all fields $F^\sigma$. It is called the normal closure of the field $F$ relative to $K$. If $L_1$ and $L_2$ are normal extensions of $K$, then so are the intersection $L_1 \cap L_2$ and the composite $L_1 \cdot L_2$. However, when $L/K'$ and $K'/K$ are normal extensions, $L/K$ 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 if and only if it is a separable extension.


[1] B.L. van der Waerden, "Algebra" , 1–2 , Springer (1967–1971) (Translated from German)
[2] S. Lang, "Algebra" , Addison-Wesley (1984)
[3] M.M. Postnikov, "Fundamentals of Galois theory" , Noordhoff (1962) (Translated from Russian)
How to Cite This Entry:
Normal extension. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by L.V. Kuz'min (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article