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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.

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Hint: Set of algebraic elements form a field over the base field.

If $(a+b)$ is algebraic, then so is $(a+b)-a=b$. Hence, a contradiction.

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  • $\begingroup$ I believe I understand that...thanks a lot! So simple. $\endgroup$ – GiGi Dec 7 '14 at 9:20

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