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

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MATH 332 - ALGEBRA AND NUMBER THEORY FIRST MIDTERM - PRACTICE TEST Note: (1) Calculators are not allowed in the exam. (2) You may assume the following axioms and theorems: (a) Axiom : The natural numbers N satisfies the Well Ordering Principle, i.e. every non-empty subset of natural numbers contains a least element. (b) 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 . Theory Problem 1. Prove that if p is prime and p | ab then either p | a or p | b . Explain why the previous statement can be re-written as follows: if p is a prime and ab 0 mod p then a 0 mod p or b 0 mod p (or equivalently, if ab 0 in Z /p Z then either a 0 or b 0 in Z /p Z ). Theory Problem 2. Prove the Fundamental Theorem of Arithmetic, i.e. every natural number n > 1 is a product of primes, and the representation is unique, except for the order of factors. Theory Problem 3. Prove Euclid’s theorem on the infinitude of primes, i.e. prove that there exist infinitely many prime numbers.
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