I am beginning to work on Dummit and Foot Section 7.4 Exercise 15 (page 257 - see attachment)

Exercise is as follows:

----------------------------------------------------------------------------------------------------------------------------------------------------------------

"Let be an element of the polynomial ring and use bar notation to denote the passage to the quotient ring .

Prove that has four elements "

---------------------------------------------------------------------------------------------------------------------------------------------------------------

Now from the result in D&F Section 7.4 Exercise 14 (see attachment) we have that every element of is of the form for some polynomial p(x) (Z/2Z)[x] of degree less than 2 ( i.e. of degree 1 or 0).

Listing such polynomials in (Z/2Z)[x] we have 0,1, x, x+1.

So the elements of are (one element in each coset!)

---------------------------------------------------------------------------------------------------------------------------------------

But what if we made the quotient ring

The same reasoning as above applies ... so (???) the elements of seem to be the same as

But this does not seem right .... is the quotient ring modulo and is the quotient ring modulo

Intuitively it seems to me that and should be different! (or maybe only the degree of the polynomial is significant?)

Can someone please clarify this matter for me?

Peter