MATH 322 Midterm: MATH 322 2010 Winter Test 1

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9 Jan 2019
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Rules governing examinations: each candidate must be prepared to produce, upon request, a. Problem 1. (3 pts) let g be a group and s g a subset. Prove that n is a subgroup of g. Problem 2. (3 pts) let f (x) z[x] be a monic polynomial, such that f ( ) = 0 for some. Problem 3. (3 pts) prove that x3 3x + 1 is irreducible in q[x]. Problem 4. (3 pts) let un zn be the multiplicative group of units. Problem 5. (3 pts) let p, q z be distinct primes. Prove that for any a z relatively prime to pq, a(p 1)(q 1) 1 (mod pq). Problem 6. (5 pts) let d2n be the dihedral group of order 2n. Let f : d12 d6 z2 be de ned by f ( i j) = ( i j, [i + j]).