View Single Post
landwolf00
#1
Sep22-10, 06:24 PM
P: 1
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
[tex]\prod [/tex] (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
[tex]\prod [/tex] (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.
Phys.Org News Partner Science news on Phys.org
Bees able to spot which flowers offer best rewards before landing
Classic Lewis Carroll character inspires new ecological model
When cooperation counts: Researchers find sperm benefit from grouping together in mice