College Algebra Exam Review 421

College Algebra Exam Review 421 - 9.4 SPLITTING FIELDS AND...

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9.4. SPLITTING FIELDS AND AUTOMORPHISMS 431 Proof. Exercise 9.4.3 . n Definition 9.4.10. A polynomial in KOExŁ is said to be separable if each of its irreducible factors has only simple roots in some (hence any) splitting field. An algebraic element a in a field extension of K is said to be sep- arable over K if its minimal polynomial is separable. An algebraic field extension L of K is said to be separable over K if each of its elements is separable over K . Remark 9.4.11. Separability is automatic if the characteristic of K is zero or if K is finite, by Theorems 9.3.1 and 9.3.4 . Theorem 9.4.12. Suppose L is a splitting field for a separable polynomial f .x/ 2 KOExŁ . Then Fix . Aut K .L// D K . Proof. Let ˇ 1 ; : : : ; ˇ r be the distinct roots of f .x/ in L . Consider the tower of fields: M 0 D K M j D K.ˇ 1 ; : : : ; ˇ j / M r D K.ˇ 1 ; : : : ; ˇ r / D L: A priori, Fix . Aut K .L// K . We have to show that if a 2 L is fixed by all elements of Aut K .L/ , then a 2 K . I claim that if a 2 M j for some j 1 , then a 2 M j 1 . It will follow from this claim that a 2 M 0 D K . Suppose that a 2 M j . If M j 1 D M j , there is nothing to show. Otherwise, let
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