By M. Pavaman Murthy, K. G. Ramanathan, C. S. Seshadri, U. Shukla, R. Sridharan
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Then Q(α)/Q is not a normal extension. e. C contains a splitting field of any polynomial in C[X]. Such fields are said to be algebraically closed. 21 Let k be a field. An irreducible polynomial f ∈ k[X] is called separable if all its roots (in the splitting field) are simple. Otherwise, f is called inseparable. A non-constant polynomial f ∈ k[X] is called separable if all its irreducible factors are separable. Let K/k be an algebraic extension. An element α ∈ K is called separable over k if the minimal polynomial of α over k is separable.
If deg f = 0 there is nothing to prove. Let deg f ≥ 1 and let f1 be an irreducible factor of f . Let α be a root of f1 and let α′ be any root of σ(f1 ). By what we have seen above, σ can be extended to an isomorphism σ1 : k(α) → k ′ (α′ ) such that σ1 (α) = α′ . Let f = (X − α)g, with g ∈ k(α)[X]. Then σ(f ) = (X − α′ )σ1 (g). The field K (resp. K ′ ) is a splitting field of the polynomial g (resp. σ1 (g) over k(α) (resp. k ′ (α′ )). By induction, σ1 admits an extension to an isomorphism τ : K → K ′ .
N respectively. Clearly, K is the splitting field of 1≤i≤n fi . 17 A normal extension K/k is an algebraic extension such that any irreducible polynomial over k which has a root in K is a product of linear factors in K. 16 that finite normal extensions are precisely splitting fields. 18 Let K/k be a finite extension. Then there exists a finite normal extension L/k such that K is a subfield of L. Let Ki /k, i = 1, 2, . . , n be finite extensions. Then there exists a finite normal extension L/k and k-isomorphisms σi of Ki into L.