A complex number $\alpha \in \mathbb{C}$ is called algebraic if there is a non-zero polynomial $P(x) \in \mathbb{Q}\left[x\right]$ with rational coefficients such that $P(\alpha)=0$. Which of the following statements is true?
- There are only finitely many algebraic numbers.
- All complex numbers are algebraic.
- $\sin \frac{\pi}{3}+ \cos \frac{\pi}{4}$ is algebraic.
- None of the above.