Unformatted text preview: n Corollary 7.5.3. Aut K .E/ acts faithfully by permutations on the roots of f.x/ in E . The action is transitive on the roots of each irreducible factor of f.x/ . Proof. By Exercise 7.4.5 , Aut K .E/ acts faithfully by permutations on the roots of f.x/ and, by the previous corollary, this action is transitive on the roots of each irreducible factor. n Theorem 7.5.4. Suppose that K ± C is a ﬁeld, f.x/ 2 KŒxŁ , and E is the splitting ﬁeld of f.x/ in C . Then Fix . Aut K .E// D K . Sketch of proof. We have a priori that K ± Fix . Aut K .E// . We must show that if a 2 L n K , then there is an automorphism of E that leaves K ﬁxed pointwise but does not ﬁx a . n Corollary 7.5.5. If K ± M ± E is any intermediate ﬁeld, then Fix . Aut M .E// D M ....
View
Full Document
 Fall '08
 EVERAGE
 Logic, Algebra, Factor Theorem, Vector Space, irreducible factor

Click to edit the document details