Question

Complete the sketch of proof for Lemma 3.17:

use theorems 3.16 and 2.5

f F is a finite dimensional separable extension of an infinite jheld Lemma 3.17. iEaa LenF. K(u) for some u ε . thern SKETCH OF PROOF. By Theorem 3.16 there is a finite dimensional Galois n field Fi of K that contains F. The Fundamental Theorem 2.5 implies that F, is finite and that the extension of K by F, has only finitely many intermediate AutA felds. Therefore, there can be only a finite number of intermediate fields in the ex- tension of K by F Since [F : K] is finite, we can choose u ε F such that IKu) :K) is maximal. If Ku) ,t F, there exists ε F-Ku). Consider all intermediate fields of the form Ku + ar) with a ε K. Since K is infinite and there are only finitely many intermediate fields, there exist a,beK such that a b and K(u at) - K(u + bc). Therefore (a-br:; (u + ar) _ (u+be) ε K(u+ar). Since a, [K(u) : K). This contra- dicts the choice of u. Hence K(u) F. Theorem 3,.16. fE is an aleebraic extension fieldoy K, then there exists an extene field F of E such that sts an extensio (i) F is normal over K; (ii) no proper subfeld of F containing (ii) ifE is separable orer K, then F is Galois over K; (iv) [F : K] is finite ifand only if[E : K] is finite. The field F is uniquely determined up to an E-isomorphism. E is normal over K Theorem 2.5. (Fundamental Theorem of Galois Theory) If F is a finite dimensional Galois extension of K, then there is a one-to-one correspondence between the set of all

Answer 1

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