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Unformatted text preview: Final Exam Solutions 1. (a) Yes, x 2 x 3 + x 2 x + 3 = x 2 x + 3 6 + x 2 x + 3 = 6 + x 2 x + 3 (b) True. Eisenstein’s criterion with p = 3 works. (c) It neither has an identity nor closure under addition. (d) [ S 5 : H ] = 120 / 10 = 12 (e) False. Consider σ = (12) ∈ S 3 . o ( σ ) = 2 and 2 6  3. (f) True. Cayley’s Theorem. (g) False. This property only holds if H is a normal subgroup of G . (h) True. Z 7 [ x ] ∼ = ≤ Z 7 ( x ) (it’s field of fractions.) (i) True. Cauchy’s Theorem. (j) True. Z 11 is a field. 2. (a) i. I is an ideal of R iff I is an additive subgroup of R and ∀ i ∈ I , ∀ r ∈ R , ir,ri ∈ I . ii. • Since I and J are ideals, they’re both nonempty and thus I + J is nonempty. • Let i 1 + j 1 and i 2 + j 2 ∈ I + J with i 1 ,i 2 ∈ I and j 1 ,j 2 ∈ J . ( i 1 + j 1 ) ( i 2 + j 2 ) = ( i 1 i 2 ) + ( j 1 j 2 . i 1 i 2 ∈ I and j 1 j 2 ∈ J since I and J are additive subgroups. Thus ( i 1 + j 1 ) ( i 2 + j 2 ) ∈ I + J . • Let r ∈ R and i 1 + j 1 ∈ I + J . r ( i 1 + j 1 ) = ri 1 + rj 1 ∈ I + J since ri 1 ∈ I and rj 1 ∈ J (because I and J are ideals. Similarly ( i 1 + j 1 ) r = i 1 r + j 1 r ∈ I + J . Thus I + J is an ideal of R . (b) i. A is the identity since Ax = xA = x for all x . ii. No. It’s not symmetric over the main diagonal....
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This note was uploaded on 04/19/2011 for the course MATH 21373 taught by Professor Johnson during the Spring '11 term at Carnegie Mellon.
 Spring '11
 Johnson
 Algebra, Addition

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