
  Let $F$ be a field. Why does $F[x]$ 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 $x^n-a$ ?

You can copy Euclid's proof. Let $p_1, \dots, p_n$ be a finite collection of prime polynomials in $F[X]$. Consider $f=p_1 p_2 \cdots p_n +1$. Let $p$ be a prime factor of $f$. Then $p$ cannot be any of $p_1, \dots, p_n$ because otherwise $p$ would divide $1$. Hence, no finite collection of prime polynomials exhausts the set of prime polynomials and so the set of prime polynomials is infinite.