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final - 1 Math 417 Final Exam August 7 2009 Solutions 1 i...

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Unformatted text preview: 1 Math 417 Final Exam: August 7, 2009: Solutions 1 . ( i ) ( 15 points ) Find gcd(924 , 125) and write it as a linear combination of 924 and 125. The Euclidean Algorithm gives (924 , 125) = 1, and 1 = 377 · 125- 51 · 924 . ( ii ) ( 10 points ) Find all solutions to the congruence 125 x ≡ 10 mod 924 . The inverse of 125 mod 924 is 377 mod 924, so that x ≡ 3770 mod 924. Therefore, the solutions are all the numbers of the form 3770+924 k for k ∈ Z . 2 . ( i ) ( 5 points ) Let R be a commutative ring. Define ideal in R and principal ideal . An ideal is a subset I ⊆ R such that 0 ∈ I ; if a,b ∈ I , then a- b ∈ I ; if a ∈ I and r ∈ R , then ra ∈ I . A principal ideal is ( a ) = { ra : r ∈ R } . ( ii ) ( 5 points ) Give an example (no proof required) of a commutative ring R and an ideal in R that is not principal. There are many examples. The ones mentioned in class are R = Z [ x ] and I the set of all polynomials with even constant term; and R = k [ x,y ], where k is a field and I = ( x,y ), the ideal generated by the indeterminants. ( iii ) ( 15 points ) If k is a field, prove that every ideal in k [ x ] is principal. This is Theorem 3.59 in the book....
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final - 1 Math 417 Final Exam August 7 2009 Solutions 1 i...

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