Past exam of the algebra 2 course of the University of Oxford / 2014 / 1 / b / Solution

Suppose, for a contradiction, that the monic irreducible polynomials in the polynomial ring $k[t]$ form a finite list $p_1,\dots ,p_r$. The list is nonempty because $t$ itself is monic and irreducible. PutThis has positive degree, so it is a nonzero nonunit. Every such polynomial over a field has an irreducible polynomial as a factor: repeatedly factor a reducible polynomial, with the positive degrees strictly decreasing until an irreducible factor is reached. Dividing that factor by its leading coefficient makes it monic; call it $q$.

By the assumed completeness of the list, $q=p_j$ for some $j$. But then $p_j$ divides both $F$ and $p_1\cdots p_r$, hence their difference $1$. A positive-degree polynomial cannot divide $1$. This contradiction proves there are infinitely many monic irreducible polynomials over every field. This is the polynomial version of Euclid's theorem; it does not require the field itself to be infinite.