$\sqrt{x}+2\sqrt{y} + 3\sqrt{z} + 4\sqrt{t}\in \mathbb{Q}$

Question

Let $x, y, z, t\in \mathbb{Q}$ s.t. $\sqrt{x}+2\sqrt{y} + 3\sqrt{z} + 4\sqrt{t}\in \mathbb{Q}$. Then show that $\sqrt{x}, \sqrt{y}, \sqrt{z}, \sqrt{t} \in\mathbb{Q}$.

Answer 1

if $sqrt(x), sqrt(y), sqrt(z), sqrt(t)$ are NOT elements of Q. Then $sqrt(x) + 2sqrt(y) + 3sqrt(z) + 4sqrt(t)$ is NOT an element of Q. If the elements are not elements of Q then they are elements of R - Q. You need to prove R - Q is closed under addition and multiplication.