Algebraic/Transcendence Theory

@DHMO At least based on this I guess that you (as the room owner) want to keep it alive. If not, just let me know, so that I do not post similar dummy posts in the future.

can you prove that $z=x+iy$ is algebraic iff $x$ and $y$ are algebraic?

8. If $a$ is a non-zero algebraic number, then $\cos(a)$ and $\sin(a)$ are both transcendental

So, $\int_0^t e^{t-x} f(x) \ \mathrm dx = e^t f(0) - f(t) + \int_0^t e^{t-x} f'(x)$

7. Let $c$ and $d$ be transcendental numbers. Then, at least one of $c+d$ and $c \times d$ is transcendental

6. Let $a$ be a complex number. $a$ and $e^a$ are linearly dependent over the rational numbers iff there exists a rational number $r$ such that $a = -W(r)$

5. If $2^t$, $3^t$, and $5^t$ are all integers, then $t$ must be an integer.

4. $e^\pi$ is transcendental

3. if $a$ is a non-zero algebraic number, then $W(a)$ is transcendental

1. $e$ is transcendental

2. $\pi$ is transcendental