Question: How can I show that the subspace $\{x\in \ell\_2 \mid \sum\_{n=1} ^\infty x\_n \frac 1{\sqrt n}=0\}$ is dense in $\ell\_2$?

Can it be done with "simple" tools just by using the definition of a dense subspace? or can it be done by using the fact that $\ell\_2$ is a Hilbert space?

Answer: 
If $f$ has a rational root, then it is an integer factor of $p$, so is
$\pm1$ or $\pm p$. Each of these can be swiftly disposed of.

There is also the possibility of $f$ factoring into two quadratic polynomials. This factorisation will have the form $(x^2+ax+1)(x^2-ax+p)$
or $(x^2+ax-1)(x^2-ax-p)$. I'll leave it to you to figure out these cases.
Split the problem into two cases: $p = 2$ and $p \neq 2$.

If $p = 2$, consider reduction modulo $3$. If $p \neq 2$, consider reduction modulo $2$. 

In both cases, it's easy to check that the polynomial has no roots over the finite field, but the possibility remains that the polynomial can factor into a product of two quadratics. However, because there are a very limited number of irreducible quadratic polynomials over $\mathbb{Z}\_2$ and $\mathbb{Z}\_3$, it is relatively easy to determine that the polynomial is ultimately irreducible over the given finite field. 

We can then conclude that the polynomial is irreducible over $\mathbb{Q}$ (see Test 3 [**here**](https://math.dartmouth.edu/archive/m31w05/public_html/Reducibility.pdf)).