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

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 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 are two distinct factorizations of...
View Full Document

This note was uploaded on 02/19/2008 for the course MATH 3320 taught by Professor Lozano-robledo during the Fall '07 term at Cornell University (Engineering School).

Page1 / 3

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online