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Unformatted text preview: University of California San Diego ECE 259A: Problem Set #3 2. (a) Prove that f x x3 x2 2 is irreducible over GF 3 . (b) Let denote the root of f x , and assume that f x is used to construct GF 27 . Compute 2 1 2 2 in GF 27 . (c) What are the possible multiplicative orders of the elements of GF 27 ? (d) What is the order of ? (b) What are all the square roots of 1 in a field of characteristic two? In a field of odd characteristic? (c) Show that if and are primitive elements in a field of odd characteristic, then is not primitive. (d) Find primitive elements and in some field of characteristic two, such that is also primitive. (b) Find a polynomial of degree 2 which is irreducible over GF 4 , and use this polynomial along with the result of (a) to construct GF 16 . (a) Show that is not primitive in the field GF 2 x (d) Find an isomorphism mapping between GF 2 x f x and GF 2 x g x . (b) Repeat (a) with "GF(2)" replaced by "GF(3)". 7. Let i , where is a primitive element of GF p m . Prove that the minimal polynomial of is M x j Ci x j , where Ci is the cyclotomic coset mod p m 1 containing i. 6. (a) Express the polynomial x 6 1 as a product of monic polynomials that are irreducible over GF 2 . How many monic polynomials divide x 6 1 over GF 2 ? (c) Find the minimal polynomial g x of (b) Show that 1 is primitive in this field. 1. 5. The polynomial f x of f x . x4 x3 x2 x 1 is irreducible over GF 2 . Let denote the root f x . 4. (a) Find a polynomial of degree 2 which is irreducible over GF 2 , and use this polynomial to construct GF 4 . 3. (a) Prove that each element of GF p m has a unique p-th root, that is p . GF p m such that 1. Let GF 29 , and let the minimal polynomial of be x 9 x 1. Find the order of . ...
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- Fall '08