sf - F has characteristic 2. 7. Let F be a eld with unity e...

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Math 3124 Thursday, December 1 Sample Final Problems since Second Test Important: the exam is comprehensive; the last third of the exam approximately will consist of material covered since the second test. Here are some typical problems. 1. Let R be a ring with exactly 3 elements { 0 , e , x } , where e is a unity for R . Prove that x 2 = e . 2. Let R be a ring and let S R . Define C = { r R | rs = sr for all s S } . Prove that C is a subring of R . 3. Let F be a field with 4 elements, and let a F with a 6 = 0 , e . (a) Prove that a 3 = e . (b) Prove that ( a - e )( a 2 + a + e ) = 0. (c) Prove that a 2 + a + e = 0. 4. Let F be a field with 9 elements. Prove that there are at least 3 homomorphisms θ : F F . Hint: freshman calculus rule. 5. Let F be a field with 3 elements. Prove that there are only 2 homomorphisms (namely the zero and identity maps) θ : F F . 6. Let F be an integral domain with 8 elements. Prove that
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Unformatted text preview: F has characteristic 2. 7. Let F be a eld with unity e , and let R be a subring of F such that e R and | R | is nite. Prove that R is an integral domain. Deduce that R is a eld. 8. Let R be a ring and let I , J R . Prove that R / I J is isomorphic to a subring of R / I R / J . Hint: dene : R R / I R / J by r = ( I + r , J + r ) . Topics covered since second test Direct sum (product) R S of rings, integral domains, elds, characteristic, freshman calculus rule, ring homomorphisms, kernel, ideals, funda-mental homomorphism theorem for rings. These are sections 24,25,26,27,38,39. Final on Friday December 9, 10:05 a.m. to 12:05 p.m. in the regular classroom (McBryde 126)....
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