
polynomial
Let $\displaystyle J $ be the set of all polynomials with zero constant term in $\displaystyle \mathbb{Z}[x] $.
(a) Show that $\displaystyle J $ is the principal ideal $\displaystyle (x) $ in $\displaystyle \mathbb{Z}[x] $.
(b) Show that $\displaystyle \mathbb{Z}[x]/J $ consists of an infinite number of distinct cosets.
So for (a) suppose it is not? For (b) you would also use contradiction?

I would think about it a little more directly. First, J is an ideal because any polynomial multiplied by an element of J will give you an element of J. It is principal because it is generated by x.
For the second part, I can't seem to give a rigourous explanation that I am happy with. Try to write a few of the cosets.
$\displaystyle \lbrace 1 + x, 2 + x, 3 + x,\ldots\rbrace$
$\displaystyle \lbrace 1 + x^2, 2 + x^2, 3 + x^2,\ldots, x+x^2, 2x+x^2, 3x+x^2 \ldots \rbrace$
They are all different and there is an infinite number of them because you can just keep raising x to a higher power. I hope I didn't confuse you with that. I'll think about it some more and try to come up with a better explanation.