Proving a subgroup is equivalent to Z

1. Feb 6, 2013

Gale

1. The problem statement, all variables and given/known data

for $n \in N, n \geq 1$ Prove that $(n^{3} +2n)Z + (n^{4}+3n^{2}+1)Z= Z$
2. Relevant 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)

3. 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!

2. Feb 6, 2013

Kreizhn

Your first idea is correct. Do you know the Euclidean algorithm and polynomial division? So that you can check your work, you should find that

$$-(n^3+2n)(n^3+2n) + (1+n^2)(n^4+3n^2+1) = 1$$

3. Feb 7, 2013

Gale

Thank you! For some reason I didn't think polynomial long division was the way to go... I'm always so unconfident when I do these sorts of proofs.

4. Feb 8, 2013

Kreizhn

There is something to be said for confidence (or perhaps intuition?). Unfortunately, it seems like a lot of the time the only way to find that confidence/intuition is to fail/succeed at a tonne of questions.

Also, something you may notice is that (like this question) there are always many ways to attack a problem. We often cannot determine which way is correct until we have followed a path through to its conclusion and found that it is a dead end. Luckily, even dead-ends often provide insight and understanding into the structure of a problem: Even through failure, we are constantly learning.