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College Algebra Exam Review 419

College Algebra Exam Review 419 - 9.4 SPLITTING FIELDS AND...

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9.4. SPLITTING FIELDS AND AUTOMORPHISMS 429 (b) j Iso K .M; L/ j D OE Aut K .L/ W Aut M .L/Ł . Proof. According to Proposition 9.4.2 , the map 7! j M is a surjection of Aut K .L/ onto Iso K .M; L/ . Check that . 1 / j M D . 2 / j M if, and only if, 1 and 2 are in the same left coset of Aut M .L/ in Aut K .L/ . This proves part (a), and part (b) follows. n Proposition 9.4.4. Let K L be a field extension and let f .x/ 2 KOExŁ . (a) If 2 Aut K .L/ , then permutes the roots of f .x/ in L . (b) If L is a splitting field of f .x/ , then Aut K .L/ acts faithfully on the roots of f in L . Furthermore, the action is transitive on the roots of each irreducible factor of f .x/ in KOExŁ . Proof. Suppose 2 Aut K .L/ , f .x/ D k 0 C k 1 x C C k n x n 2 KOExŁ; and ˛ is a root of f .x/ in L . Then f . .˛// D k 0 C k 1 .˛/ C C k n n // D .k 0 C k 1 ˛ C C k n ˛ n / D 0: Thus, .˛/ is also a root of f .x/ . If A is the set of distinct roots of f .x/ in L , then 7! j A is an action of Aut K .L/ on A . If L is a splitting field for f .x/ , then, in particular, L D K.A/ , so the action of Aut K .L/ on A is faithful. Proposition 9.4.1 says that if
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