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Exam 1 Modern Algebra II, Dave Bayer, October 2, 2008 Name: [1] (6 pts) [2] (6 pts) [3] (6 pts) [4] (6 pts) [5] (6 pts) TOTAL Please work only one problem per page, starting with the pages provided. Clearly label your answer. If a problem continues on a new page, clearly state this fact on both the old and the new pages. [1] Define a ring homomorphism. Define an ideal. Prove that the kernel of a ring homomorphism is an ideal. [2] Let A be an n × n matrix with entries in R, satisfying the polynomial relation (x − 2)3 = 0 Find a formula for eAt as a polynomial expression in A. Give an example of a matrix A for which this is the minimal polynomial relation, and check your formula using this matrix. [3] Construct the finite field F8 as an extension of F2 = Z/2Z, by finding an irreducible polynomial of degree 3 with coefficients in F2 . What are the three roots of your irreducible polynomial? [4] A message is represented as an integer a mod 35. You receive the encrypted message a5 ≡ 3 mod 35. What is a? [5] Give an example of a finite ring R which is not a field, such that 1 6= 0 but 1 + 1 + 1 = 0.
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