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Unformatted text preview: PRACTICE FINAL EXAM FOR MAT 312 AND AMS 351 (1) (a) Use the Euclidean algorithm to compute gcd ( a,b ) for the integers a = 124 and b = 38. (b) Find integers c,d such that gcd ( a,b ) = ca + db . (b) Is 38 invertible in Z 124 ? (c) Is the ring Z 124 a field? (2) (a) Use the Euclidean algorithm to compute gcd ( a ( x ) ,b ( x )) for the poly- nomials a ( x ) = x 5 +4 x 2- x +4 and b ( x ) = x 6- x 2 in the polynomial ring R [ x ] where R =real numbers. (b) Is a ( x ) invertible in the quotient ring R [ x ] /b ( x )? (c) Is the quotient ring R [ x ] /b ( x ) a field? (3) Solve the equation  n [ x ] n =  n , for n = 263. (This is a congruence problem mod n.) (4) Let G denote a group satisfying | G | = 23, i.e. G contains 23 members. Explain why G must be an abelian group. (5) Let G denote a group satisfying | G | = 24, and let g G . Explain why g 73 = g . (6) State Lagranges Theorem. State Burnsides Theorem....
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