4023f11ps5

# 4023f11ps5 - Z 2 . Which of these are irreducible? 5....

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Homework #5 Due: October 5, 2011 1. Compute the product (5 x 3 + 4 x 2 + 3)(6 x 2 + 3 x + 5) in Z 7 [ x ]. 2. Let p ( x ) = x 3 + x 2 + 1 2 Z 2 [ x ], and let K = Z 2 [ x ] = ( p ( x )). If t = x 2 K , then K can be represented by K = ' a 0 + a 1 t + a 2 t 2 : a 0 ; a 1 ; a 2 2 Z 2 ; where t satisﬂes the relation t 3 + t 2 + 1. (See Theorem 0.8.4.) Find the multiplicative inverse of each of the following elements of K . Express your answers in the above form: a 0 + a 1 t + a 2 t 2 . (a) t (b) t + 1 (c) t 2 + t + 1 3. List all of the polynomials of degree 2 over Z 2 . Which of these are irreducible? 4. List all of the polynomials of degree 3 over
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Unformatted text preview: Z 2 . Which of these are irreducible? 5. Nagpaul-Jain, Page 46, #3. 6. Show that p ( x ) = x 4 + x 3 + x 2 + x + 1 is irreducible over Z 2 , but is not a primitive polynomial. Hint: Problem 3 may be useful. Also, compute x 5 in Z 2 [ x ] = ( p ( x )). 7. Nagpaul-Jain, Page 46, #7. Write down the multiplication table for your eld. Math 4023 1...
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