Separable Extension - Separable Extensions Within Algebraic Extensions

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

Famous quotes containing the words extensions and/or algebraic:

    The psychological umbilical cord is more difficult to cut than the real one. We experience our children as extensions of ourselves, and we feel as though their behavior is an expression of something within us...instead of an expression of something in them. We see in our children our own reflection, and when we don’t like what we see, we feel angry at the reflection.
    Elaine Heffner (20th century)

    I have no scheme about it,—no designs on men at all; and, if I had, my mode would be to tempt them with the fruit, and not with the manure. To what end do I lead a simple life at all, pray? That I may teach others to simplify their lives?—and so all our lives be simplified merely, like an algebraic formula? Or not, rather, that I may make use of the ground I have cleared, to live more worthily and profitably?
    Henry David Thoreau (1817–1862)