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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!

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    Numerator means the polynomials in indeterminate $x$ with coefficients in GF(2). The /(x^3-1) means "mod x^3-1" i.e. two pol$y$nomials are considered the same if they give the same remainder on division by x^3-1.2011-04-20

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$\mbox{GF}(2)$ 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 $x^3-1$.

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    3. No, of course not! $(x^3-1)$ is the ideal generated by $x^3-1$.2011-04-20