[Noisebridge-discuss] Help with finite fields.

[Sai Emrys]

On Thu, Dec 3, 2009 at 11:03 PM, Ben Kovitz <bkovitz at indiana.edu> wrote:
> The notation R[x] means "the ring of polynomials in x".  That is, all
> the functions of the form ax^n + bx^(n-1) + ... + cx + d, where a, b,
> c, d are all elements of R, and (typically) so is x.  In the place
> where you found it, (Z/2Z)[T] *really* means (I'm translating loosely)
> polynomials where the coefficients are 0 or 1 and the x's are
> integers.  (Z/2Z)[T]/(T^2 + T + 1) means the factor ring resulting
> from treating T^2 + T + 1 as the 'zero' of that ring of polynomials.

I think it's confused to say that x is typically an element of R, or T
an integer.  Really you want to treat x as just a formal symbol, just
some device to make the addition and multiplication of these lists (a,
b, c, d, ...) of coefficients in R behave as you want them to, and not
an element of something preexisting.
Of course x _will_ be an element of R[x] (and there are evaluation
maps from R[x] to R where you plug in an element of R for x), but if x
were actually an element of R to begin with the usual interpretation
of the notation would just make R[x] = R, since you're not adding
anything new to the ring.

- Alex again