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Prelim 1 Solutions

Prelim 1 Solutions - MATH 332 ALGEBRA AND NUMBER THEORY...

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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 . ———————– 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. Proof. Assume that all numbers have at least one factorization into primes and let S be the set of all natural numbers which have two distinct factorizations. Then either S is empty and we would be done, or S is a non-empty set of natural numbers. By the Well Ordering Principle, the set S has a least element n and so there are two distinct factorizations: n = p a 1 1 · · · p a r r = q b 1 1 · · · q b s s . But then p 1 divides q b 1 1 · · · q b s s and so, by Theorem (b) above, there is some i such that p 1 divides q b i i (assume i = 1) and therefore p 1 divides q 1 . Since they are both primes they should be the same prime, hence p 1 = q 1 . But then: n p 1 = p a 1 - 1 1 · · · p a r r = q b 1 - 1 1 · · · q b t t
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