What is the problem in proving $\pi$ is transcendental over $\mathbb{Q}$ in purely algebraic method

Question

Why can’t we prove for the field extension $\mathbb{R}$ over $\mathbb{Q}$ ,$\pi$ is transcendental over $\mathbb{Q}$ in a purely algebraic method? Or is there any prove that proves $\pi$ is transcendental over $\mathbb{Q}$ in a purely algebraic method?