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I recently found a linear algebra question from the very beginning of my first semester. Without any knowledge of abstract algebra, we were supposed to prove that if $p$ and $q$ are distinct primes, the numbers $1$, $\sqrt{p}$ and $\sqrt{q}$ are $\mathbb{Q}$-linearly independent. I could not solve it back then and still can't now.

I really want to do it the brute force way without any background knowledge (maybe except $\sqrt{p}, \sqrt{q} \notin \mathbb{Q}$), i.e. prove that if $$ \lambda_1 + \lambda_2\sqrt{p}+\lambda_3 \sqrt{q}= 0 $$ for some $\lambda_1, \lambda_2, \lambda_3 \in \mathbb{Q}$, then $\lambda_1 = \lambda_2 = \lambda_3 = 0$.

I tried to move $\lambda_3 \sqrt{q}$ to the RHS and square both sides, but without much success.

Hyperbolic PDE friend
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1 Answers1

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Put $\lambda_1$ to the other side, then square, which gives you that $\lambda_2\lambda_3\sqrt{pq}$ is rational. Hence one of $\lambda_2,\lambda_3$ is zero.

daw
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