Quotient Rings of Polynomials Over Finite Fields

Question

I have this question which I don't know how to approach:

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Let ${F}_{2} = {Z}/2Z$, find representatives for the residue classes of ${F}_{2}[X]$ modulo the polynomial $f(x)$ and compute the multipication table for the ring ${F}_{2}[X]/(f(x))$ where

1. $f(x) = x+1$
2. $f(x) = x^2+x+1$
3. $f(x) = x^2+1$

which are those rings are fields?

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I understand $F_2$ and ${F}_{2}[X]$ but I don't know what the residues are and how to compute the multiplication table or how multiplication is defined in this ring. I suppose that checking if something is a field is easy once you have the multiplications table.

Answer 1

Do you know how you can define multiplication $ab$ in the ring ${\bf Z}/7{\bf Z}$ by multiplying $a$ times $b$ as integers, dividing by 7, and reporting the remainder as the answer? Well, it's the same thing: if $a$ and $b$ are in ${\bf F}_2[X]$ then you multiply them as polynomials, divide by $f(x)$, and report the remainder as the answer. For example, if $f(x)=x^2+x+1$, and you want to multiply $x$ by $x$, you get $x^2$, then divide by $x^2+x+1$, and the remainder is $x+1$ (well, $-x-1$, but, over ${\bf F}_2$, it's the same thing), so that's the answer.

Answer 2

let $f=ax^n+\ldots+a_0$ where $a_i\in F$ and $a_n\neq 0$ then we have $F[x]/(f(x))=\{b_{n-1}x^{n-1}+\ldots +b_0\}$

Now you are in $F_2$ so how many choices we have for $b_{n-1}$,...,$b_0$ ?