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MA554_AUG05

# MA554_AUG05 - QUALIFYING EXAMINATION AUGUST 2005 MATH 554...

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QUALIFYING EXAMINATION AUGUST 2005 MATH 554 - Dr. C. Wilkerson There are eight problems, each worth 25 points for a total of 200 points.Unless otherwise stated, show all necessary work. All rings are assumed to be commutative rings with a multiplicative iden- tity element. I. (a) Let A be a ﬁnite abelian group of order 9 * 256. Let φ n : A A be the group homo- morphism that sends x nx , for any integer n . The following information is known about ker( φ n ) n #ker( φ n ) #ker( φ 2 n ) #ker( φ 3 n ) 2 8 64 256 3 3 9 9 Deduce the structure of A as a direct sum of cyclic groups of prime power order. Give the in- variant factors for A . (b) Let V be an 8 dimensional vector space over a ﬁeld K and let ψ End K ( V ). Suppose that the kernel of ( ψ - 5) j has dimension k over K and that the following is known about k : for j =1 , k = 4; for j =2 , k = 7, and for j =3 , k = 8. Write down the rational canonical form and Jordan canonical form for ψ . II. (a) Deﬁne the concepts of Euclidean domain, PID, and UFD.

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MA554_AUG05 - QUALIFYING EXAMINATION AUGUST 2005 MATH 554...

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