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Concept
Separable extension
Summary
In field theory, a branch of algebra, an algebraic field extension E/F is called a separable extension if for every \alpha\in E, the minimal polynomial of \alpha 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 f
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