Let u a algebraic number and v a transcendent number, so any rational linear combination of u and v will also be transcendent. How to prove it?
2026-03-26 13:00:51.1774530051
How to prove that any rational linear combination of u and v will also be transcendent?
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Let $x:=\alpha u+\beta v$ with $\alpha,\beta$ rational (and $\beta\not=0$). If $x$ is not transcendental, it would be algebraic. But then also $v=\frac1\beta(x-\alpha u)$ is algebraic, a contradiction.