4023f11ex2a

4023f11ex2a - Name: Solutions Exam 2 Instructions. Answer...

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Name: Solutions 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 . I Solution. Over Z 3 , p (0) = 1, p (1) = 2, and p (2) = 2. Thus, p ( x ) has no roots in Z 3 , and since deg p ( x ) = 2 it follows that p ( x ) is irreducible. Over Z 5 , p (2) = 2 2 +1 = 0 so x 2 +1 = ( x ¡ 2)( x ¡ 3) 2 Z 5 [ x ] and hence is reducible. J (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. I Solution. j F j = 3 2 = 9. F = f 0 ; 1 ; 2 ; t; t + 1 ; t + 2 ; 2 t; 2 t + 1 ; 2 t + 2 g . J 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 . I Solution. z 2 = ( t + 1) 2 = t 2 + 2 t + 1 = ¡ 1 + 2 t + 1 = 2 t , z 4 = ( z 2 ) 2 = (2 t ) 2 = t 2 = ¡ 1 = 2, and 1 z = 1 t + 1 = 1 t + 1
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4023f11ex2a - Name: Solutions Exam 2 Instructions. Answer...

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