Explain how to obtain a field with $125$ elements using polynomials by using concrete example provided by a polynomial

Question

I have this problem and don't know how to solve it. Any one can help? Thanks

Answer 1

Take $\mathbb{F}_5[x] / (p(x))$ where $p(x)$ is an irreducible polynomial of degree $3$.

One example of an irreducible polynomial of degree $3$ in $\mathbb{F}_5[x]$ is $x^3 + x + 1$. We can check this by evaluating that $x^3 + x + 1$ has no roots.

$\mathbb{F}_5 [x]$ is a PID since $\mathbb{F}_5$ is a field. Since $p(x)$ is irreducible in $\mathbb{F}_5[x]$, $\mathbb{F}_5[x] / (p(x))$ is a field.