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math 311w finalexam

math 311w finalexam - Questions for the exam 1 Theoretical...

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Questions for the exam April 27, 2010 1 Theoretical questions 1. Formulate binomial theorem. Right the formula for ( x + y ) n and formula for the binomial coefficients. 2. For two positive integer numbers n, m define gcd ( n, m ) and prove that it exists. You can not use prime number factorization, you can use well ordering principle. 3. For two polynomials n ( x ) , m ( x ) define gcd ( n ( x ) , m ( x )) and prove that it exists. 4. Prove that if integers numbers a and c are relatively prime and c divides ab then c divides b . 5. Prove that if polynomials a ( x ) and c ( x ) are relatively prime and c ( x ) divides a ( x ) b ( x ) then c ( x ) divides b ( x ). 6. Prove that there are infinitely many prime numbers. 7. Prove that if a product of several numbers is divisible by a prime number p then one of them is divisible by p . 8. Prove that if a product of several polynomials is divisible by an irreducible polynomial p ( x ) then one of them is divisible by p ( x ). 9. a)Prove that any positive integer number could be written as a product of prime numbers n = p 1 p 2 · · · p k b) Prove that the factorization is unique in the sense that if also n = q 1 q 2 · · · q l then l = k and we can renumber the q i , so that q i = p i . 10. What polynomials are irreducible over R ? You don’t need to prove it, only describe them so it was easy to check if polynomial is irreducible or not.

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