Rationality of $a^2+b^2$

Question

I have looked into this topic lately and have not found an answer to the following question. Is the following true:
If $a,b\in\mathbb{R}$ and $a + b$ is rational, then $a^2 + b^2$ is rational

Answer 1

No.

Proof by counter-example

Take any irrational number $c$ and let $a = \sqrt c$ and $b = -\sqrt c$.

Then $a+b$ will be rational, but $a^2 + b^2$ will be irrational.