Let be the set of all polynomials with zero constant term in .

(a) Show that is the principal ideal in .

(b) Show that consists of an infinite number of distinct cosets.

So for (a) suppose it is not? For (b) you would also use contradiction?

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- Oct 12th 2009, 04:52 AMSampraspolynomial
Let be the set of all polynomials with zero constant term in .

(a) Show that is the principal ideal in .

(b) Show that consists of an infinite number of distinct cosets.

So for (a) suppose it is not? For (b) you would also use contradiction? - Oct 12th 2009, 09:15 AMeeyore
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.

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.