Irrationality of sum of roots of primes.

[caffeinemachine]

I observed the following:

1) $\sqrt{2}$ is irrational.

2) $\sqrt{2}+\sqrt{3}$ is irrational(since its square is irrational).

3) $\sqrt{2}+\sqrt{3}+\sqrt{5}$ is irrational(assume its rational and is equal to $r$. Write $r- \sqrt{5}=\sqrt{2} + \sqrt{3}$. Now square both the sides and its obvious from here).

So I am thinking may be $\sqrt{p_1} + \sqrt{p_2} + \sqrt{p_3} + \ldots + \sqrt{p_n}$ is irrational for all $n$, where $p_i$ is the $i-th$ prime.

The trick I used for (3) doesn't work for $n=4$. Any ideas anyone?

 

[QuestForInsight]

The square roots of the primes are linearly independent over the field of rationals.

 

[caffeinemachine]

The square roots of the primes are linearly independent over the field of rationals.

Thank You!