First Prelim - MATH 332 ALGEBRA AND NUMBER THEORY FIRST...

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MATH 332 - ALGEBRA AND NUMBER THEORY - FIRST MIDTERM - 10/02/2007 Show all work. No calculators. There are 2 theory questions and 4 additional problems. You may assume the following axioms and theorems: (1) Axiom : The natural numbers N satisfy the Well Ordering Principle, i.e. every non- empty subset of natural numbers contains a least element. (2) Theorem: Let a, b, c be integers. The linear equation ax + by = c has a solution if and only if gcd( a, b ) divides c . (3) Theorem: Let p be a prime and let a, b be any integers. If p | ab then p | a or p | b . More generally, if p | a 1 a 2 · · · a k then p divides some a i . Just in case you need them, the following are all the primes below 100: 2 , 3 , 5 , 7 , 11 , 13 , 17 , 19 , 23 , 29 , 31 , 37 , 41 , 43 , 47 , 53 , 59 , 61 , 67 , 71 , 73 , 79 , 83 , 89 , 97 . ———————– Theory Question 1. (20 points) Prove the uniqueness part of the Fundamental Theo- rem of Arithmetic , i.e. assume that every natural number n has a factorization and then prove that this factorization is unique up to the order of the factors.
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