{\displaystyle E\supseteq F} If A separable extension is an extension that may be generated by separable elements, that is elements whose minimal polynomials are separable. i = , = However, irreducibility depends on the ambient field, and a polynomial may be irreducible over F and reducible over some extension of F. Similarly, divisibility by a square depends on the ambient field. Let f in F[X] be an irreducible polynomial and f' its formal derivative. } ⊇   denotes the tensor product of fields, F {\displaystyle X^{p}-a} F Then there exists an embedding ˙0: K!Ls.t. The proofs of Theorems 1.1 and 1.2 both use tensor products. a Suppose that such an intermediate extension does exist, and [E : F] is finite, then [S : F] = [E : K], where S is the separable closure of F in E.[18] The known proofs of this equality use the fact that if Then jAut(K=F)j [K: F], and equality holds if and only if K=F is Galois. >> {\displaystyle \alpha \in E} α {\displaystyle E\supseteq F} Let X /Filter /FlateDecode   is separable over F if it is algebraic over F, and its minimal polynomial is separable (the minimal polynomial of an element is necessarily irreducible). p   the E-vector space of the F-linear derivations of E, one has. E   is separable algebraic over x over F is not a separable polynomial, or, equivalently, for every element x of E, there is a positive integer k such that For instance, the polynomial g(X) = X2 – 1 has precisely deg(g) = 2 roots in the complex plane; namely 1 and –1, and hence does have distinct roots. ⊇ ⊇ A separating transcendence basis of an extension such that for every A finitely generated field extension is separable if and only it has a separating transcendence basis; an extension that is not finitely generated is called separable if every finitely generated subextension has a separating transcendence basis.[21]. {\displaystyle a_{i}=b_{i}^{p}} is injective. F E , ∈ t&��M�����m��SO4Mv�d��FGhQ�gj��_h���МC쫖� h�k�����z���*�F@�;�9GcRvoL)c1�6���`��q%`/3� �g��P��t��,)�����7T|(� �g�~8���9��O8>M�&\�"(L�q����i�}��� f&�f��̄Q��np\��m�+}.�� _��1�� �3�����`��x�DZo�c0!�":m5�:� Űu@U(���e���g�w;�F�L�]v�(g�A�E�$�Q�fI~�����i�c�������z2eT 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 supposed to be separable.