[Noisebridge-discuss] Help with finite fields.

[Ben Kovitz]

On Fri, Dec 4, 2009 at 2:20 AM, Sai Emrys <noisebridge at saizai.com> wrote:
> 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.

I agree.  I was trying to keep that previous message from getting any
more abstract and confusing, cut a corner, and ended up making it even
more confusing.  Maybe I was confused by the fact that the very little
bit I've done with these things involved evaluation homomorphisms,
where we were indeed ultimately evaluating the polynomials.

Anyway, thanks for the correction.

Ben