Inspired by Jason Chrysoprase

I have had problems with mycomputer, like always. Let p be a prime number your question is relationed with the cyclotomical polynomials... For each p a prime number, I'm going to give you a proposition and later and example

Proposition.- Let $F$ be a field with caractheristic(p) $\neq 0$ , for example, $\mathbb{Z}_p$ , then are equivalents:

a) F admits a cyclic extension of degree p.

b) There exists $a \in F$ such that $x^p - x - a$ is irreducible over $F$

c) There exists $a \in F$ such that $\beta^p - \beta \neq a, \forall \beta \in F$ .

Example.-

a) All extension $F \subset E$ over finite fields is a cyclic extension.

b) The monic irreducible polynomials of degree 4 in $\mathbb{Z}_2[x]$ are:

b1) $x^4 + x^3 + x^2 + x + 1$

b2) $x^4 + x^3 + 1$

b3) $x^4 + x + 1$

Realize that there is not whole classification of groups under isomorphisms... There is only one whole classification of groups under isomorphims for finitely generated abelian groups... But I hope to be able to answer your question the best possible... For example, in $\mathbb{R}[x]$ the irreducible polynomials has degree 1 or at most 2... in $\mathbb{Q}[x]$ the irreducible polynomials are able to have any degree ( think about the cyclotomic polynomials, Eisenstein's criterion...)