polynomials - sum of square root of primes 2

I dont know how to solve the problem below.

(1) $p[1]$, $p[2]$, $\ldots$, $p[n]$ are distinct primes, where $n = 1,2,\ldots$
Let $a[n]$ be the sum of square root of those primes, that is, $a[n] = \sqrt{p[1]}+\ldots+\sqrt{p[n]}$. Show that there exists a polynomial with integer coefficients that has $a[n]$ as a solution.

(2) Show that $a[n]$ is irrational.

I can solve this without using (1) for (2). But my teacher said (1) is a hint for (2).
Help please?