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Concept
Separable extension
Summary
In field theory, a branch of algebra, an algebraic field extension is called a separable extension if for every , the minimal polynomial of over F is a separable polynomial (i.e., its formal derivative is not the zero polynomial, or equivalently it has no repeated roots in any extension field). There is also a more general definition that applies when E is not necessarily algebraic over F. An extension that is not separable is said to be inseparable.
Every algebraic extension of a field of characteristic zero is separable, and every algebraic extension of a finite field is separable.
It follows that most extensions that are considered in mathematics are separable. Nevertheless, the concept of separability is important, as the existence of inseparable extensions is the main obstacle for extending many theorems proved in characteristic zero to non-zero characteristic. For example, the fundamental theorem of Galois theory is a theorem about normal extensions, which remains true in non-zero characteristic only if the extensions are also assumed to be separable.
The opposite concept, a purely inseparable extension, also occurs naturally, as every algebraic extension may be decomposed uniquely as a purely inseparable extension of a separable extension. An algebraic extension of fields of non-zero characteristics p is a purely inseparable extension if and only if for every , the minimal polynomial of over F is not a separable polynomial, or, equivalently, for every element x of E, there is a positive integer k such that .
The simplest example of a (purely) inseparable extension is , fields of rational functions in the indeterminate x with coefficients in the finite field . The element has minimal polynomial , having and a p-fold multiple root, as . This is a simple algebraic extension of degree p, as , but it is not a normal extension since the Galois group is trivial.
An arbitrary polynomial f with coefficients in some field F is said to have distinct roots or to be square-free if it has deg f roots in some extension field .
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