1. ## Another field contruction

Construct a field of order 125.

I don't know, this chapter is just not coming to me for some reasons, I can't really understand how to construct a field. So I pick a field, then pick a poly that is irreducible and has order of 125, how do I do that?

Construct a field of order 125.

I don't know, this chapter is just not coming to me for some reasons, I can't really understand how to construct a field. So I pick a field, then pick a poly that is irreducible and has order of 125, how do I do that?
Again remember what I said. Let $F$ be a finite field $\mathbb{Z}_p$. Let $p(x) = a_nx^n+...+a_1x+a_0$ be an irreducible polynomial over $F$. Then $F/\left< p(x) \right>$ is a field. Now any element in $F/\left< p(x) \right>$ can be written uniquely as $b_0+b_1x+...+b_{n-1}x^{n-1}$ now for $b_i$ (for $0\leq i \leq n-1$) we have $p$ choices for the coefficients. Thus, in total there are $p^n$ such elements in this larger field.

1)Let $F=\mathbb{Z}_5$
2)Let $p(x)$ be an order $3$ irreducible polynomial in $F$.
3)The factor ring $F/\left< p(x)\right>$ is a field with $5^3=125$ elements.

So the thing remaining now is for you to find an irreducible degree polynomial.

3. $(x-2)^3$, it has zero in there. would that be it?

$(x-2)^3$, it has zero in there. would that be it?