Hello!

If we take

to be the ring of polynomials with integer coefficients, and

to be the subset

.

So I is the ring of polynomials with even integer coefficients

So far, I have shown that I is an ideal.

I am trying to

(i) find & describe the number of cosets in

(ii) Prove that

is not a principle ideal of

(iii) Decide if

is a prime ideal.

Well, for (i), I am thinking there is infinitely many cosets? As long as any

has coefficient 1, it forms it's own coset... for example

etc... in terms of coset equality... That seems to be it to me...

, which is true...Note that an element

in the quotient ring is uniquely determined by its free coefficient's parity...
(ii) I am not quite too sure how to go about this. If we let

for some

and derive a contradiction (show we can't just take multiples of p?)

First show that , and then suppose . Check that this

is impossible by noting that both and checking the possible degree of
(iii) I think this will be easy once (i) is figured out... If

is an integral domain, then

is prime. If

is not integral domain, then

is not prime.

Indeed, this follows at once from (i)

Tonio
Any help appreciated! Thanks in advance!