Exercise 2.24

Proof. Notations :

$\mathcal{}P$ : set of all monic polynomials $p$ in $k[x]$.

${\mathcal{}P}_n$ : set of all monic polynomials $p$ in $k[x]$ with $\mathrm{deg}<!--\; FUNCTION\; APPLICATION\; -->(p)\le n$.

$\mathcal{}M$ : set of all monic irreducible polynomials $p$ in $k[x]$.

${\mathcal{}M}_n$ : set of all monic irreducible polynomials $p$ in $k[x]$ with $\mathrm{deg}<!--\; FUNCTION\; APPLICATION\; -->(p)\le n$.

We must prove that $\underset{p\in \mathcal{}M}{\sum <!--\; FUNCTION\; APPLICATION\; -->}{q}^{-\mathrm{deg}<!--\; FUNCTION\; APPLICATION\; -->p(x)}$ diverges.

$\text{∙}$ $\underset{p\in \mathcal{}P}{\sum <!--\; FUNCTION\; APPLICATION\; -->}{q}^{-\mathrm{deg}<!--\; FUNCTION\; APPLICATION\; -->p(x)}$ diverges :

So $\underset{f\in \mathcal{}P}{\sum <!--\; FUNCTION\; APPLICATION\; -->}{q}^{-\mathrm{deg}<!--\; FUNCTION\; APPLICATION\; -->f}$ diverges.

$\text{∙}$ $\underset{f\in \mathcal{}P}{\sum <!--\; FUNCTION\; APPLICATION\; -->}{q}^{-2\mathrm{deg}<!--\; FUNCTION\; APPLICATION\; -->f}$ converges :

As any finite subset of $\mathcal{}P$ is included in some ${\mathcal{}P}_n$, $\underset{f\in \mathcal{}P}{\sum <!--\; FUNCTION\; APPLICATION\; -->}{q}^{-2\mathrm{deg}<!--\; FUNCTION\; APPLICATION\; -->f}$ converges.

$\text{∙}$ $\underset{p\in \mathcal{}M}{\sum <!--\; FUNCTION\; APPLICATION\; -->}{q}^{-\mathrm{deg}<!--\; FUNCTION\; APPLICATION\; -->p(x)}$ diverges :

Let ${\mathcal{}M}_n=\{p_1,p_2,\dots ,p_{l(n)}\}$ the set of all monic irreducible polynomials such that $\mathrm{deg}<!--\; FUNCTION\; APPLICATION\; -->p_i\le n$. Let

For simplicity, we write $l=l(n)$ for a fixed $n\in \mathbb{N}$. Then

Since the monic prime factors of any polynomial $p\in {\mathcal{}P}_n$ are in ${\mathcal{}P}_n$, the decomposition of $p$ is $p=p_1^{a_1}\cdots p_l^{a_l}$, so

So $\underset{n\to \infty }{\lim <!--\; FUNCTION\; APPLICATION\; -->}\lambda (n)=\infty$ : this is another proof that there exist infinitely many monic irreducible polynomials in $k[x]$ (cf Ex. 2.1).

Yet

(the last inequality is equivalent to $2\le q^{\mathrm{deg}<!--\; FUNCTION\; APPLICATION\; -->p_i}$). So

As $\frac{1}{q^{2\mathrm{deg}<!--\; FUNCTION\; APPLICATION\; -->p_1}}+\cdots +\frac{1}{q^{2\mathrm{deg}<!--\; FUNCTION\; APPLICATION\; -->p_{l(n)}}}$ is less than the constant $\underset{f\in \mathcal{}P}{\sum <!--\; FUNCTION\; APPLICATION\; -->}{q}^{-2\mathrm{deg}<!--\; FUNCTION\; APPLICATION\; -->f}$, if $\underset{p\in \mathcal{}M}{\sum <!--\; FUNCTION\; APPLICATION\; -->}{q}^{-\mathrm{deg}<!--\; FUNCTION\; APPLICATION\; -->p(x)}$ converges, then $\log <!--\; FUNCTION\; APPLICATION\; -->\lambda (n)\le C$, where $C$ is a constant, so $\lambda (n)\le e^C$ for all $n\in \mathbb{N}$, in contradiction with $\underset{n\to \infty }{\lim <!--\; FUNCTION\; APPLICATION\; -->}\lambda (n)=\infty$.

Conclusion : $\underset{p\in \mathcal{}M}{\sum <!--\; FUNCTION\; APPLICATION\; -->}{q}^{-\mathrm{deg}<!--\; FUNCTION\; APPLICATION\; -->p(x)}$ diverges. □