Quotient rings - help!

From: N.C. (leo1476_at_hotmail.com)
Date: 09/06/04 

Date: 6 Sep 2004 11:13:24 -0700

Hi,

I could not find any algebra book that really defines quotient rings
in polynomials. I know the elements of quotient rings are all the
left/right cosets, but how do they look like? I wish some book would
make this clear. It is so frustrating.

If R[x] is a polynomial ring, and I[x] is some ideal of R[x], how do
the elements of R[x]/I[x] look like?

For example, I can't visualize these:

a. Z[x]/(2, x) for instance; or

b. Z[x]/(x); or

c. Q[x]/(y - x^2); or

d. Z[x]/(2, x^2-x+1). How do I show that this is a finite field? How
many elements does it have? How can I show maximality of (2, x^2-x-1)?
or finiteness of the quotient?

Thanks a lot in advance.

N.C.

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