Let F be a field F[x] denotes the ring of polynomials in one variable over the field.
Why does this ring have infinitely many irreducible elements?
For the case F has characteristic 0
Then x-a is irreducible for all a $\in F$ since x satisfies no non-trivial relations in F.
Obviously this argument fails for a finite field since there are only finitely many a to choose from.
So how may I construct irreducible polynomials in a finite field?
I figure it must involve higher powers of x, maybe $x^n-a$ ?