College Algebra Exam Review 424

College Algebra Exam Review 424 - Mx . 9.5. The Galois...

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434 9. FIELD EXTENSIONS – SECOND LOOK using Corollary 9.4.3 , Proposition 9.4.16 , and the multiplicativity of di- mension, Proposition 7.3.1 . n Corollary 9.4.18. Let K ± M be a finite–dimensional separable field extension. Then j Aut K .M/ j ² dim K M: Proof. There is a field extension K ± M ± L such that L is finite– dimensional and Galois over K . (In fact, M is obtained from K by adjoin- ing finitely many separable algebraic elements; let L be a splitting field of the product of the minimal polynomials over K of these finitely many elements.) Now, we have j Aut K .M/ j ² j Iso K .M;L/ j D dim K M . n Exercises 9.4 9.4.1. Prove Proposition 9.4.7 . 9.4.2. Prove Proposition 9.4.8 . 9.4.3. Prove Proposition 9.4.9 . 9.4.4. Suppose that f.x/ 2 KŒxŁ is separable, and K ± M is an extension field. Show that f is also separable when considered as an element in
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Unformatted text preview: Mx . 9.5. The Galois Correspondence In this section, we establish the fundamental theorem of Galois theory, a correspondence between intermediate elds K M L and subgroups of Aut K .L/ , when L is a Galois eld extension of K . Proposition 9.5.1. Suppose K L is a nitedimensional separable eld extension. Then there is an element 2 L such that L D K./ . Proof. If K is nite, then the nitedimensional eld extension L is also a nite eld. According to Corollary 3.6.26 , the multiplicative group of units of L is cyclic. Then L D K./ , where is a generator of the multiplicative group of units....
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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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