Normal extension
of a field
An algebraic field extension (cf. Extension of a field) of
satisfying one of the following equivalent conditions:
1) any imbedding of in the algebraic closure
of
comes from an automorphism of
;
2) is the splitting field of some family of polynomials with coefficients in
(cf. Splitting field of a polynomial);
3) any polynomial with coefficients in
, irreducible over
and having a root in
, splits in
into linear factors.
For every algebraic extension there is a maximal intermediate subfield
that is normal over
; this is the field
, where
ranges over all imbeddings of
in
. There is also a unique minimal normal extension of
containing
. This is the composite of all fields
. It is called the normal closure of the field
relative to
. If
and
are normal extensions of
, then so are the intersection
and the composite
. However, when
and
are normal extensions,
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 separable (cf. Separable extension).
References
[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) |
Normal extension. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Normal_extension&oldid=38571