subject: abstract algebra

topic: divisibility

Please show all work and justify each step!

(a) Let (1,1) 6 Z and p E N prime. Let q,'r be the quotient and remainder for the division of a by b in Z. Assume p | a and p | b. Prove that: p | 7. Prove your claim. (Make sure to formally spell out every single step of your proof.) (b) Consider the constant polynomials 12 and 5 in QM. What is their greatest common divisor? Prove your claim. (c) Let F be a eld and f (23), g(:c) E F[a:]. Assume f (as) and g(:c) are nonassociate and both irreducible. Prove that f (as) and g(:c) are relatively prime