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sp2005 - F such that for every x F either x R or x-1 R...

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ALGEBRA COMPREHENSIVE EXAMINATION Bishop* Spring 2005 Brookfield Cates. Answer 5 questions only. You must answer at least one from each of groups, rings, and fields. Be sure to show enough work that your answers are adequately supported. GROUPS 1. Let G be an abelian group, H = { a 2 * a 0 G } and K = { a 0 G * a 2 = e }. Prove that H G/K . 2. Assume G =HZ ( G ) , where H is a subgroup of G and Z ( G ) is the center of G . Show: a. Z ( H ) = H 1 Z ( G ) b. G ' = H ' (Where G ' is the derived group of G ) c. G/Z ( G ) H/Z ( H ) 3. Prove: a. A group of order 80 need not be abelian (twice) by exhibiting two non-isomorphic non- abelian groups of order 80 (with verification). b. A group of order 80 must be solvable. RINGS 1. Let R be a ring with ideals A and B . a. Define a natural function n : R/A 1 B 6 R/A × R/B and show that it is a ring homomorphism. b. Calculate Ker( n ), the kernel of n . c. Prove that if R = A + B , then n is an isomorphism. d. Show that the converse of (c) is false. 2. Let
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Unformatted text preview: F such that, for every x F , either x R or x-1 R . Prove that the ideals of R are linearly ordered; i.e., if I and J are ideals of R , then either I f J or J f I . 3. Let M 2 ( Q ) be the ring of 2×2 all matrices with rational enties. Prove: a. M 2 ( Q ) has no nontrivial ideals. b. M 2 ( Q ) has an identity but is not a field. FIELDS 1. Find the minimal polynomial for " = over the field of rationals Q and prove it is minimal. 5 2 + 2. Let GF( p n ) denote the Galois field with p n elements. (a) Prove that GF( p a ) f GF( p b ) iff a divides b . (b) Prove that GF( p a ) 1 GF( p b ) = GF( p d ), where d = gcd( a, b ). 3. Let F be a finite field of n = p m elements. Find necessary and sufficient conditions to insure that f ( x ) = x 2 + 1 has a root in F ; i.e., f is not irreducible over F ....
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