The following example showed up in my textbook, and I am struggling to understand a certain part of it.
Theorem 17.5 simply states that if $F$ is a field and $f(x)$ exists in $F[x]$, and is of degree 2 or 3, then $f(x)$ is reducible if and only if $f(x)$ has a zero in $F$. The corollary then says $F[x]/<f(x)>$ is a field.
The part I do not follow is:
$Z_2[x]/<x^3 + x + 1>$ = { $ax^2 + bx + c + <x^3 + x + 1> | a,b,c \epsilon Z_2$ }
I understand that $x^3 + x + 1$ is irreducible over $Z_2[x]$ and therefore it is a field(directly from the theorem and corollary), and I understand that given the equality, the field has 8 elements, but I do not understand the equailty itself.
I am still getting a grasp of quotient rings and polynomial rings so I apologize in advance if I don't understand something that is rather obvious.
Thanks for any help.
edit: Perhaps I do not fully understand the reasoning behind picking a cubic polynomial either. But I think it's because if we picked a quadratic polynomial, then our quotient ring wouldn't contain(all?) quadratic polynomials, and our field would be too small, is that correct?