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## Homework Statement

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.

## Homework Equations

First isomorphism theorem

## 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 :)