My question is given some set of transcendental numbers can we using algebraic operations form an algebraic number? My intuitive answer is no, could you please tell me what branch of mathematics it is so that I can search for rigorous proof? Or maybe you can answer my question on your own. Thanks!

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The answer is :

Yes sometimes if you have more than one transcendental numbers, or if you're willing to do stuff like $x-x$. For the first part, assume you're given $\{a,b\}$ where $a= \pi$, $b=\pi^2$. Then $a^2 - b$ gives you an algebraic number. The second part is obvious.

No if you have only one transcendental number and are not willing to do stuff like $x-x$. Or if you have more than one transcendental number and if they're algebraically independent (which basically means exactly what you were asking for).

The theory you're looking at is field theory, more specifically the theory of field extensions (what you're calling algebraic operations in this context are compositions of elementary field operations)

Maxime Ramzi

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