Lectures on the Algebraic Theory of Fields - Tata Institute of ...

4. Separability 25For, the minimum polynomial ofωover L divides that over k.2) ω∈Ω separable over k⇔k(ω)/k has inΩ(k(ω) : k) distinct k- 29isomorphisms.Let now K= k(ω 1 ,...,ω n ) and letω 1 ,...ω n be all separable over k.Put K i = k(ω 1 ,...,ω) so that K o = k and K n = K. Now K i−1 (ω i ) andω i is separable over K i−1 so that K i over K i−1 has exactly (K i : K i−1 )distinct K i−1 - isomorphisms. This proves that K has over k(K n : K n−1 )...(K 1 : K o )=(K n : K o )=(K : k)distinct k-isomorphisms. If thereforeω∈K, Then by previous articlek(ω) has exactly (k(ω) : k) distinct isomorphisms and henceωis separable over k. Conversely if K/k is finite and every elementof K is separable over k, then K/k has exactly (K : k) distinct k-isomorphisms. Hence3) (K : k) < ∞, K/k has (K : k) distinct k-isomorphisms⇔everyelement of K is separable over k.Let us now make theDefinition. A subfield K ofΩ/k is said to be separable over k if everyelement of K is separable over k.From 3) and the definition, it follows that4) K/k is separable⇔ for every subfield L of K with (L : K)