College Algebra Exam Review 417

College Algebra - 9.4 SPLITTING FIELDS AND AUTOMORPHISMS 427 9.3.7 If K is a field of characteristic p à a 2 K then.x C a/p D x p C  and p Hint

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 9.4. SPLITTING FIELDS AND AUTOMORPHISMS 427 9.3.7. If K is a field of characteristic p à a 2 K , then .x C a/p D x p C  and p . Hint: The binomial coefficient p is divisible by p if 0 < k < p . a k Now, suppose K is a field of characteristic p and that f .x/ is an irreducible polynomial in KŒx. If f .x/ has a multiple root in some extension field, then Df .x/ is identically zero, by Exercise 9.3.6. Therefore, there is a g.x/ 2 KŒx such that f .x/ D g.x p / D a0 C a1 x p C : : : ar x rp . p Suppose that for each ai there is a bi 2 K such that bi D ai . Then f .x/ D .b0 C b1 x C br x r /p , which contradicts the irreducibility of f .x/. This proves the following theorem: Theorem 9.3.2. Suppose K is a field of characteristic p in which each element has a p t h root. Then any irreducible polynomial in KŒx has only simple roots in any field extension. Proposition 9.3.3. Suppose K is a field of characteristic p . The map a 7! ap is a field isomorphism of K into itself. If K is a finite field, then a 7! ap is an automorphism of K . Proof. Clearly, .ab/p D ap b p for a; b 2 K . But also .a C b/p D ap C b p by Exercise 9.3.7. Therefore, the map is a homomorphism. The homomorphism is not identically zero, since 1p D 1; since K is simple, the homomorphism must, therefore, be injective. If K is finite, an injective map is bijective. I Corollary 9.3.4. Suppose K is a finite field. Then any irreducible polynomial in KŒx has only simple roots in any field extension. Proof. K must have some prime characteristic p . By Proposition 9.3.3, any element of K has a p t h root in K and, therefore, the result follows from Theorem 9.3.2. I 9.4. Splitting Fields and Automorphisms Recall that an automorphism of a field L is a field isomorphism of L onto L, and that the set of all automorphisms of L forms a group denoted by ...
View Full Document

This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.

Ask a homework question - tutors are online