College Algebra Exam Review 329

College Algebra Exam Review 329 - 7.5. SPLITTING FIELDS OF...

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Unformatted text preview: 7.5. SPLITTING FIELDS OF POLYNOMIALS IN C Œx 339 Proof. E is also the splitting field of f over M , so the previous result applies with K replaced by M . I Now, we consider a converse: Proposition 7.5.6. Suppose K  E  C are fields, dimK .E/ is finite, and Fix.AutK .E// D K . (a) For any ˇ 2 E , ˇ is algebraic over K , and the minimal polynomial for ˇ over K splits in EŒx. (b) For ˇ 2 E , let ˇ D ˇ1 ; : : : ; ˇr be a list of the distinct elements of f .ˇ/ W 2 AutK .E/g. Then .x ˇ1 /.: : : /.x ˇr / is the minimal polynomial for ˇ over K . (c) E is the splitting field of a polynomial in KŒx. Proof. Since dimK .E/ is finite, E is algebraic over K . Let ˇ 2 E , and let p.x/ denote the minimal polynomial of ˇ over K . Let ˇ D ˇ1 ; : : : ; ˇr be the distinct elements of f .ˇ/ W 2 AutK .E/g. Define g.x/ D .x ˇ1 /.: : : /.x ˇr / 2 EŒx. Every 2 AutK .E/ leaves g.x/ invariant, so the coefficients of g.x/ lie in Fix.AutK .E// D K . Since ˇ is a root of g.x/, it follows that p.x/ divides g.x/. On the other hand, every root of g.x/ is of the form .ˇ/ for 2 AutK .E/ and, therefore, is also a root of p.x/. Since the roots of g.x/ Q simple (i.e., each root ˛ occurs only once in the factorization are g.x/ D .x ˛ /), it follows that g.x/ divides p.x/. Hence p.x/ D g.x/. In particular, p.x/ splits into linear factors over E . This proves parts (a) and (b). Since E is finite–dimensional over K , it is generated over K by finitely many algebraic elements ˛1 ; : : : ; ˛s . It follows from part (a) that E is the splitting field of f D f1 f2 fs , where fi is the minimal polynomial of ˛i over K . I Definition 7.5.7. A finite–dimensional field extension K  E  C is said to be Galois if Fix.AutK .E// D K . With this terminology, the previous results say the following: Theorem 7.5.8. For fields K  E  C , with dimK .E/ finite, the following are equivalent: (a) The extension E is Galois over K . ...
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