# Linear independence in $\mathbb{R}$ [duplicate]

I would like to prove that the family $\{\sqrt{p}, p\text{ prime number} \}$ is linearly independent in $\mathbb R$ where $\mathbb R$ is a $\mathbb Q$-vector space.

I know how to prove this for up to 4 elements but I would like a general proof as elementary as possible.

• Thanks, so it seems there is not a very elementary solution. This exercise comes from a list of exercises for beginning linear algebra and the next exercise asked the same question for $\log(p)$ (that is quite easy). – Sebastian Aug 21 '12 at 17:04