Exercise 2.1

Proof. Suppose that the set $S$ of irreducible polynomials is finite : $S=\{P_1,P_2,\dots ,P_n\}$.

Let $Q=P_1{P}_2\cdots P_n+1$. As $S$ contains the polynomials $x-a,a\in k$, $\mathrm{deg}<!--\; FUNCTION\; APPLICATION\; -->(Q)\ge q=|k|>1$. Thus $Q$ is divisible by an irreducible polynomial. As $S$ contains all the irreducible polynomials, there exists $i,1\le i\le n$, such that $P_i\mid Q=P_1{P}_2\cdots P_n+1$, so $P_i\mid 1$, and $P_i$ is an unit, in contradiction with the irreducibility of $P_i$.

Conclusion: $k[x]$ has infinitely many irreducible polynomials. As each polynomial has only a finite number of associates, there exist infinitely many monic irreducible polynomials. □