(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

I have to show that [tex]\sum[/tex] ai x^{i}-> (a0 [tex]\sum[/tex] ai) is a ring homomorphism from C[x] to C x C

I then have to use the first isomorphism theorem to show that there is an isomorphism from C[x]/ (x(x-1)) to C x C where (x(x-1)) is the principal ideal (p) generated by the element p=x(x-1) of C[x]

It then asks is C[x]/(x(x-1)) an integral domain.

2. Relevant equations

First isomorphism theorem

3. The attempt at a solution

I think I may have done the first part but I'm a little unsure if my notation/understanding is fully correct, considering multiplication can I say f(([tex]\sum[/tex] ai x^{i})([tex]\sum[/tex] bj x^{j}) = f ([tex]\sum[/tex] (aibj) x^{k}) {summing over i+j=k} = (a0,[tex]\sum[/tex]ai)(bo,[tex]\sum[/tex]bj) = f([tex]\sum[/tex] ai x^{i}) f([tex]\sum[/tex] bj x^{j})

I'm more stuck on the second part. I think I am aiming to show that ker f = (x(x-1))

ker f = {a0 + a1x + ... + anx^{n}: a0 = 0, a1 + ... +an = 0}

= {x (a1 + ... + anx^{n-1}) : a1+ ... +an =0}

Now I thought about trying to pull out a factor of x-1 but I don't think that would work?

I can see that x(x-1) does satisfy that there is no constant term and the sum of ai is 0 but I don't see how to get that all polynomials multiplied by it do.

Thanks :)

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: First isomorphism theorem for rings

**Physics Forums | Science Articles, Homework Help, Discussion**