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: 103:00 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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