{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Math 114 - Spring 1994 - Bergman - Midterm 2

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

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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. ...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online