meaning of $GF(2)[x]/(x^3-1)$ What does $GF(2)[x]/(x^3-1)$ mean? I know $GF(2)$ is the Galois field with 2 elements, but what does the forward slash mean? Also, what's the meaning of the entire expression?
Thanks!
 A: $\mbox{GF}(2)$ is the finite field with 2 elements (one of the rare instances in mathematics where the common name for an object is kind of larger than the object itself). 
$\mbox{GF}(2)[x]$ is the ring of polynomials in the variable $x$ with coefficients in that field. If you're not sure what this means, you should probably learn about this first before you tackle the expression at hand. Let's call this ring $R$.
$x^3-1$ is a specific polynomial in that ring. Since the field is $\mbox{GF}(2)$, it's actually the same polynomial as $x^3+1$. 
$(x^3-1)$ is the ideal generated by that polynomial: it's all the polynomials of the form $(x^3-1)f(x)$ where $f(x)$ is an arbitrary polynomial over $\mbox{GF}(2)$. Let's call this ideal $I$.
The forward slash means "quotient" - the quotient of ring $R$ by the ideal $I$, denoted $R/I$. One good way to think about it is as the the set of polynomials of degree 2 (or less), endowed with the operations of ordinary polynomial addition and ordinary multiplication except that after multiplying you take the remainder of the result upon division by $x^3-1$. 
