Let $a$ and $b$ be elements in extension field $F$. Is it true that:
If $a$ is algebraic over $F$ and $b$ is transcendental over $F$, then $a+b$ is transcendental over $F$?
I just did the same problem with both $a$ and $b$ as transcendental and it was easy to see that $a+b$ isn't transcendental when $a=\pi$ and $b=-\pi$. I've been trying to find another smooth counterexample for this instance but I've come to think that algebraic $+$ transcendental $=$ transcendental.
I want to assume $a+b$ is algebraic and create a contradiction showing that $b$ can then not be transcendental. I don't have any ideas how I can lead myself to that contradiction however.