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Math 114 - Spring 1994 - Bergman - Midterm 2

Math 114 - Spring 1994 - Bergman - Midterm 2 - SUN 15:45...

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Unformatted text preview: 11/11/2001 SUN 15:45 FAX 6434330 MOFFITT LIBRARY .001 George M. Bergman Spring 1994, Math 114 8 April, 1994 41 Evans Hall Second Midterm Exam 2: 10-3:00 PM 1. (30 points) Let n be a positive integer, K a field in which the polynomial In — l splits, and a an element of a field extension of K such that Dane K. Prove that 1((05) :K is normal, and that 1"(K(oc) :K) is abelian. 2. (20 points) Let p be a fixed prime. (a) (10 points) If G is a finite group, define what is meant by a Sylow p—subgroup of G. (b) (10 points) We have proved that every finite group G has a Sylow p—subgroup, and that any two Sylow p—subgroups of G are conjugate. What statements do these results yield about subextensions of a finite separable normal field extension L: K, on applying the Fundamental Theorem of Galois Theory? (Statements only; no proofs or arguments required.) 3. (20 points) Let G be a simple group, and d an integer >1 such that G has an element of order d. Show that G is generated by the set X of all its elements of order 0’. 4. (30 points) In all three parts of this problem, assume L: K is a field extension, and Ca, fiEL are two elements each of which generates the extension: [C(05) = L = K03). (a) (8 points) Show that if or is algebraic over K, then B is also algebraic, and of the same degree. (b) (8 points) Show by example that or and [3 can be algebraic with different minimal polynomials over K. (c) (14 points) Prove that if or is separable over K, then so is [3. ...
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