Use of irreducible polynomial in finite field construction

Question

When constructing a finite field $\mathrm{GF}(p^n)$ using polynomials:

1. Why do we need to modulo an irreducible polynomial? What happens if this polynomial is reducible?

2. Does such an irreducible polynomial always exist? If so, is there a systematic way to find it when $n$ is large?

Answer 1

If $f$ is not irreducible then $k[x]/(f)$ is not an integral domain, if it is irreducible then $k[x]/(f)$ is a field. Depending on the field $k$ only some degrees of irreducible $f$ might exist, but for $k$ a finite field there are irreducible polynomials of every degrees, this is because the splitting field of $x^{q^n}-x\in k[x]$ is a field with $q^n$ elements, where $q = cardinality(k)$. There is no simple way to find some $f$ of a given large degree.