Separable extension
In mathematics, a separable extension of a field K is a field L containing K that can be generated by adjoining to K a set of elements α, each of which is a root of a separable polynomial over K. In that case, each β in L has a separable minimal polynomial over K.
The condition of separability is central in Galois theory. Since though all fields of characteristic 0, and all finite fields, are perfect (all extensions separable), the condition can be assumed in many contexts.
The effects of inseparability (necessarily for infinite K of characteristic p) can be seen in the primitive element theorem, and for the tensor product of fields.
Given a finite extension L/K of fields, there is a smallest subfield M of L containing K such that L is a separable extension of M. When L = M the extension L/K is called a purely inseparable extension. In general finite extensions factor as a separable extension of a purely inseparable extension, because a separable extension of a separable extension is again a separable extension.
Referenced By
List of abstract algebra topics | List of mathematical topics (S-U)
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