- Separable extension
In
mathematics , analgebraic field extension "L"/"K" is separable if it can be generated by adjoining to "K" a set each of whose elements is a root of aseparable polynomial over "K". In that case, each β in "L" has a separableminimal polynomial over "K".The condition of separability is central in
Galois theory . A perfect field is one for which all finite (equivalently, algebraic) extensions are separable. There exists a simple criterion for perfectness: a field "F" is perfect if and only if*"F" has characteristic 0, or
*"F" has a nonzero characteristic "p", and every element of "F" has a "p"-th root in "F".Equivalently, the second condition says that the
Frobenius endomorphism of "F", xmapsto x^p, is anautomorphism .In particular, all fields of characteristic 0 and all
finite field s are perfect. This means that the separability condition can be assumed in many contexts. The effects of inseparability (necessarily for infinite "K" of characteristic "p") can be seen in theprimitive element theorem , and for thetensor product of fields .Given a finite extension "L"/"K" of fields, there is a largest subfield "M" of "L" containing "K" such that "M" is a separable extension of "K". When "M" = "K" the extension "L"/"K" is called a purely inseparable extension. In general an algebraic extension factors as a purely inseparable extension of a separable extension, since the compositum of a family of separable extensions is again separable.
Purely inseparable extensions do occur for quite natural reasons, for example in
algebraic geometry in characteristic "p". If "K" is a field of characteristic "p", and "V" analgebraic variety over "K" of dimension > 0, consider thefunction field "K"("V") and itssubfield "K"("V")"p" of "p"-th powers. This is always a purely inseparable extension. Such extensions occur as soon as one looks at multiplication by "p" on anelliptic curve over a finite field of characteristic "p".In dealing with non-perfect fields "K", one introduces the separable closure "K"sep inside an
algebraic closure , which is the largest separable subextension of "K"alg/"K". Then Galois theory can be carried out inside "K"sep.References
*cite book|last=Hungerford |first=Thomas |title=Algebra |year=1974 |publisher=Springer |id=ISBN 0-387-90518-9
* | year=2002 | volume=211
*cite book |last=Silverman |first=Joeseph |title=The Arithmetic of Elliptic Curves |year=1993 |publisher=Springer |id=ISBN 0-387-96203-4
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