4023f11ex2

4023f11ex2 - Name: Exam 2 Instructions. Answer each of the...

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Name: Exam 2 Instructions. Answer each of the questions on your own paper, and be sure to show your work so that partial credit can be adequately assessed. Put your name on each page of your paper. 1. [ 26 Points] (a) Verify that p ( x ) = x 2 +1 is irreducible over the fleld Z 3 , but it is reducible over the fleld Z 5 . (b) If F = Z 3 [ x ] = ( x 2 + 1) then F can be represented by F = f a + bt : a; b 2 Z 3 g ; where t is the congruence class of x . F is a fleld since x 2 + 1 is irreducible over Z 3 . i. How many elements are there in the fleld F . List all of them. ii. Let z = t + 1 2 F . Compute z 2 , z 4 , and 1 =z in the fleld F . Express each of your answers in the standard form a + bt for some a; b 2 Z 3 . iii. Verify that t is not a generator of the multiplicative group F / , but that z = t + 1 is a generator of F / . Your calculations in part ii may be useful. 2. [ 40 Points] Let H = 2 4 0 1 1 1 1 0 0 1 0 1 1 0 1 0 1 1 0 1 0 0 1 3 5 be a matrix with entries in Z 2 . (a) Assume
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