Prove that $a$ and $b$ are rational numbers

Question

If $a+b$, $a^2+b$ and $b^2+a$ are rational numbers and $a+b\neq 1$ then $a$ and $b$ are rational.

I try, sum the expresions but I only got that $a^2+b^2$ and $ab$ are rational.

Any suggestion?

Answer 1

Write : $a^2 + b - (b^2 + a) =(a-b)(a+b-1)$ , this is rational. Since $a+b$ is rational, so is $a+b-1$. It is non-zero, therefore $a-b$ is rational being the quotient of two rationals, the latter non-zero.

Of course, if $a+b$ and $a-b$ are rationals, so are $a$ and $b$.

Answer 2

Hint: Since $a^2+b$ and $a+b$ are both rational, $a^2-a$ is rational, which means that $(2a-1)^2$ is rational. Similarly $(2b-1)^2$ is rational. Now,

$$(2a-1)+(2b-1)=2(a+b-1)$$

is a nonzero rational. Can you finish from here?