# Finite fields and products of polynomials

1. Sep 22, 2010

### landwolf00

1. The problem statement, all variables and given/known data
This question is in two parts and is about the field F with q = p^n for some prime p.
1) Prove that the product of all monic polynomials of degree m in F is equal to
$$\prod$$ (x^(q^n)-x^(q^i), where the product is taken from i=0 to i=m-1
2) Prove that the least common multiple of all monic polynomials of degree m in F is equal to
$$\prod$$ (x^(q^i)-x)[/tex], where the product is taken from i=1 to i=m

2. Relevant equations
N/A

3. The attempt at a solution
I did an induction argument on part 1 of the problem, which i believe to be correct. All polynomials of degree m+1 are representable uniquely as x*f+a, where f has degree m, and a is an element of Fq. there is probably a better solution, and i'm not even sure how to start the second part of the problem.