College Algebra Exam Review 421

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

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Unformatted text preview: 9.4. SPLITTING FIELDS AND AUTOMORPHISMS 431 I Proof. Exercise 9.4.3. Definition 9.4.10. A polynomial in KŒx 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 separable 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 KŒx. Then Fix.AutK .L// D K . Proof. Let ˇ1 ; : : : ; ˇr be the distinct roots of f .x/ in L. Consider the tower of fields: M0 D K   Mj D K.ˇ1 ; : : : ; ˇj /   Mr D K.ˇ1 ; : : : ; ˇr / D L: A priori, Fix.AutK .L// à K . We have to show that if a 2 L is fixed by all elements of AutK .L/, then a 2 K . I claim that if a 2 Mj for some j 1, then a 2 Mj 1 . It will follow from this claim that a 2 M0 D K . Suppose that a 2 Mj . If Mj 1 D Mj , there is nothing to show. Otherwise, let ` > 1 denote the degree of the minimal polynomial p.x/ ` for ˇj in Mj 1 Œx. Then f1; ˇj ; ; ˇj 1 g is a basis for Mj over Mj 1 . In particular, ` a D m0 C m1 ˇj C C m` 1 ˇj 1 (9.4.1) for certain mi 2 Mj 1 . Since p.x/ is a factor of f .x/ in Mj 1 Œx, p is separable, and the ` distinct roots f˛1 D ˇj ; ˛2 ; : : : ; ˛` g of p.x/ lie in L. According to Proposition 9.4.1, for each s , there is a s 2 AutMj 1 .L/  AutK .L/ such that s .˛1 / D ˛s . Applying s to the expression for a and taking into account that a and the mi are fixed by s , we get a D m0 C m1 ˛s C C m` `1 1 ˛s (9.4.2) for 1 Ä s Ä `. Thus, the polynomial .m0 a/ C m1 x C C m` 1 x l 1 of degree no more than ` 1 has at least ` distinct roots in L, and, therefore, the coefficients are identically zero. In particular, a D m0 2 Mj 1 . I ...
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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.

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