Separable extension
of a field
An extension such that for some natural number
the fields
and
are linearly disjoint over
(see Linearly-disjoint extensions). An extension that is not separable is called inseparable. Here
is the characteristic of
. In characteristic 0 all extensions are separable.
In what follows only algebraic extensions will be considered (for transcendental separable extensions see Transcendental extension). A finite extension is separable if and only if the trace mapping is a non-zero function. An algebraic extension is separable if any finite subextension is separable.
The separable extensions form a distinguished class of extensions, that is, in a tower of fields the extension
is separable if and only if
and
are separable; if
and
are separable extensions, then so is
; for a separable extension
and an arbitrary extension
the extension
is again separable. An extension
is separable if and only if it admits an imbedding in a Galois extension
. In this case, the number of different
-isomorphisms of
into
is the same as the degree
for a finite extension
. Any finite separable extension is simple.
A polynomial is called separable over
if none of its irreducible factors has a multiple root in an algebraic closure of
. An algebraic element
is called separable (over
) if it is a root of a polynomial that is separable over
. Otherwise
is called inseparable. An element
is called purely inseparable over
if
for some
. An irreducible polynomial
is inseparable if and only if its derivative
is identically zero (this is possible only for
of characteristic
and
). An arbitrary irreducible polynomial
can be uniquely represented in the form
, where
is a separable polynomial. The degree of
and the number
are called, respectively, the reduced degree and the index of
.
Let be an arbitrary algebraic extension. The elements of the field
that are separable over
form a field,
, which is the maximal separable extension of
contained in
. The field
is called the separable closure of
in
. The degree
is called the separable degree of
, and the degree
the inseparable degree, or the degree of inseparability. The inseparable degree is equal to some power of the number
. If
, then
is said to be separably closed in
. In this case the extension
is called purely inseparable. An extension
is purely inseparable if and only if
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that is, if any element of is purely inseparable over
. The purely inseparable extensions of a field
form a distinguished class of extensions. If an extension
is both separable and purely inseparable, then
. For references see Extension of a field.
Separable extension. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Separable_extension&oldid=21154