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Separable extension

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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Related concepts (17)
Splitting field
In abstract algebra, a splitting field of a polynomial with coefficients in a field is the smallest field extension of that field over which the polynomial splits, i.e., decomposes into linear factors. A splitting field of a polynomial p(X) over a field K is a field extension L of K over which p factors into linear factors where and for each we have with ai not necessarily distinct and such that the roots ai generate L over K. The extension L is then an extension of minimal degree over K in which p splits.
Separable polynomial
In mathematics, a polynomial P(X) over a given field K is separable if its roots are distinct in an algebraic closure of K, that is, the number of distinct roots is equal to the degree of the polynomial. This concept is closely related to square-free polynomial. If K is a perfect field then the two concepts coincide. In general, P(X) is separable if and only if it is square-free over any field that contains K, which holds if and only if P(X) is coprime to its formal derivative D P(X).
Galois extension
In mathematics, a Galois extension is an algebraic field extension E/F that is normal and separable; or equivalently, E/F is algebraic, and the field fixed by the automorphism group Aut(E/F) is precisely the base field F. The significance of being a Galois extension is that the extension has a Galois group and obeys the fundamental theorem of Galois theory. A result of Emil Artin allows one to construct Galois extensions as follows: If E is a given field, and G is a finite group of automorphisms of E with fixed field F, then E/F is a Galois extension.
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