Proof of the Algebraic Independence of π and γ

This paper establishes a criterion for the algebraic independence of two numbers. Applying this theorem, we prove that π and the Euler–Mascheroni constant γ are algebraically independent. Consequently, it follows that πγ, π/γ, π+γ , and π−γ are all transcendental. Furthermore, we deduce that at least one of the following statements must hold: π and e are algebraically independent, or γ and e are algebraically independent.