- #1

Gale

- 684

- 2

## Homework Statement

for [itex] n \in N, n \geq 1 [/itex] Prove that [itex] (n^{3} +2n)Z + (n^{4}+3n^{2}+1)Z= Z[/itex]

## Homework Equations

I know subgroups of Z are of the form aZ for some a in Z and also that aZ+bZ= dZ, where d=gcd(a,b)

## The Attempt at a Solution

So I was thinking if I could prove that the gcd of (n^3+2n) and (n^4+3n^2+1) was 1, then I could make the proof, but I'm struggling to figure out how to find a gcd of two polynomials... I also tried factoring to see if that led anywhere, but it didn't really...

Then I was thinking that if I could show that 1 was in the group, and since 1 generates Z, that would prove that the group was equivalent to Z... but then I wasn't actually sure that logic was sound.

Any help or some guidance in the right direction would be appreciated. Thanks!