Checking subspaces for independence

Question

Assume: $V:= \mathbb{Q}[\sqrt{2}] = \left \{ a+b \sqrt{2} \right \}$ My task is to check if the following subset is linear independent: $\\$

$\left \{ 10, 3+ \sqrt{2} \right \}$

My approach: $\alpha 10 + \beta ({3+ \sqrt{2}})=0$.
I am looking for a non trivial solution. Therefore I am searching for at least one facor $\alpha$ , $\beta$ such that one term cancels out and I can set the other factor to zero therefore the hole equation is zero. But I have no idea how to solve it..

Answer 1

Hint:

Try to express that square root. Remember that all elementary operations are closed in the set of rational numbers.

Solution:

So we have $$\beta \sqrt{2} = -10\alpha -3\beta$$ Now if $\beta \ne 0$ then $$ \sqrt{2} = {-10\alpha -3\beta\over \beta}$$ which i nonsense since LHS is irational and RHS is not. So $\beta =0$...