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

Exercise is as follows:

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"Let $\displaystyle x^2 +x + 1$ be an element of the polynomial ring $\displaystyle E= {\mathbb{F}}_2 [x] $ and use bar notation to denote the passage to the quotient ring $\displaystyle \overline{E} = {\mathbb{F}}_2 [x]/<x^2 + x + 1>$.

Prove that $\displaystyle \overline{E} $ has four elements $\displaystyle \overline{0}, \overline{1}, \overline{x}, \overline{x+1} $"

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Now from the result in D&F Section 7.4 Exercise 14 (see attachment) we have that every element of $\displaystyle \overline{E} $ is of the form $\displaystyle \overline{p(x)} $ for some polynomial p(x) $\displaystyle \in $ (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 $\displaystyle \overline{E} $ are $\displaystyle \overline{0}, \overline{1}, \overline{x}, \overline{x+1} $ (one element in each coset!)

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But what if we made the quotient ring $\displaystyle \overline{G} = {\mathbb{F}}_2 / <x^2 +1> $

The same reasoning as above applies ... so (???) the elements of $\displaystyle \overline{G} $ seem to be the same as $\displaystyle \overline{E} $

But this does not seem right .... $\displaystyle \overline{E} $ is the quotient ring modulo $\displaystyle x^2 + x + 1 $ and $\displaystyle \overline{G} $ is the quotient ring modulo $\displaystyle x^2 + 1$

Intuitively it seems to me that $\displaystyle \overline{E} $ and $\displaystyle \overline{G} $ should be different! (or maybe only the degree of the polynomial is significant?)

Can someone please clarify this matter for me?

Peter