Separable Extensions Within Algebraic Extensions
Separable extensions occur quite naturally within arbitrary algebraic field extensions. More specifically, if is an algebraic extension and if, then S is the unique intermediate field that is separable over F and over which E is purely inseparable. If is a finite degree extension, the degree is referred to as the separable part of the degree of the extension (or the separable degree of E/F), and is often denoted by sep or s. The inseparable degree of E/F is the quotient of the degree by the separable degree. When the characteristic of F is p > 0, it is a power of p. Since the extension is separable if and only if, it follows that for separable extensions, =sep, and conversely. If is not separable (i.e., inseparable), then sep is necessarily a non-trivial divisor of, and the quotient is necessarily a power of the characteristic of F.
On the other hand, an arbitrary algebraic extension may not possess an intermediate extension K that is purely inseparable over F and over which E is separable (however, such an intermediate extension does exist when is a finite degree normal extension (in this case, K can be the fixed field of the Galois group of E over F)). If such an intermediate extension does exist, and if is finite, then if S is defined as in the previous paragraph, sep==. One known proof of this result depends on the primitive element theorem, but there does exist a proof of this result independent of the primitive element theorem (both proofs use the fact that if is a purely inseparable extension, and if f in F is a separable irreducible polynomial, then f remains irreducible in K). The equality above (sep==) may be used to prove that if is such that is finite, then sep=sepsep.
If F is any field, the separable closure Fsep of F is the field of all elements in an algebraic closure of F that are separable over F. This is the maximal Galois extension of F. By definition, F is perfect if and only if its separable and algebraic closures coincide (in particular, the notion of a separable closure is only interesting for imperfect fields).
Read more about this topic: Separable Extension
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