[Math] Determine and construct irreducible polynomial (of degree >3) over finite fields.

Question

Wanted to determine whether a polynomial $x^p+…+a_0$ was irreducible.

The common sense came from that

A polynomial of degree two or three over a field F was irreducible if and only if it had a root in $F$.

and

A monic polynomial with integer coefficients and $p(d)\neq0$ for all integer dividing the constant term of $p(x)$, then $p(x)$ had no root in $Q$.

However, the first one only limited in polynomial of degree 2 or 3. The second one, like Eisenstein's Criterion and many others, was in $\mathbb{Z}[x]$ or UFDs.

I also read an article in Construction of Irreducible Polynomials over Finite Fields and the procedure was way too messy, and the remaining option seemed to actually calculated out all the possible outcomes.

My question was that:

1. Was there any theorems to determine polynomials irreducible or not in a finite field? (Especially, monic polynomials of degree 4 -6.)

2. Was there any other general procedure to create irreducible polynomials?

Answer 1

There are a few general algorithms for factoring polynomials over a finite field; see Wikipedia.

To find all irreducible polynomials of degree $n$ over $\mathbb F_p$, factor $x^{p^n}-x$ mod $p$. See an example here.