OK, I am asked to find the elements for the addition and multiplication tables for the finite fields with eight elements. I know I've asked about this question previously, and I've almost gotten a complete understanding of the problem. This is what I've done:
I know that we cannot construct our finite field with a composite, but rather the prime version of it:
$$\mathbb{F_8}=\mathbb{F_{2^3}}$$.
Instead, we reduce the field by finding an isomorphic field to it. We can do this by picking ANY irreducible polynomial:
$$\mathbb{F_{2^3}}\cong \mathbb{F_2}[x]/(x^3+x^2+1)$$
This is where I get stuck. What are the elements of this isomorphic field? If I can get that then I can figure out exactly what the multiplication and addition tables are which is what the problem is asking for.
Someone mentioned that it would be all the polynomials in $\mathbb{Z_2}$ lower than degree 3. Is this true? If so, or not, what would be the best way to find the required polynomials to solve the problem? Thanks so much for the help thus far!
Answer
The correct isomorphism is $$ \mathbb{F}_8\cong \mathbb{F}_2[x]/(x^3+x+1), $$ which contains (by definition of a quotient ring) the equivalence classes of the eight polynomials $$ 0,1,x,1+x,x^2,1+x^2,1+x+x^2,x+x^2. $$ Classes of higher degree polynomials reduce to these elements, since $x^3=x+1=-(x+1)$ in $\mathbb{F}_2[x]$.
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