While proving that the character values in $\mathbb{C}$ of a finite group are algebraic integers, one usually proceeds as follows:
(1) Prove a very general theorem in algebraic number theory - algebraic integers are closed under addition and multiplication.
(2) For any character $\chi$ of a finite group $G$, $\chi(g)$ is sum of certain roots of unity in $\mathbb{C}$ (depending on $o(g)$).
(3) Since roots of unity are algebraic integers, so is their sum.
Question: Since the character values are sums of very specific algebraic integers: those $\alpha \in\mathbb{C}$ satisfying $\alpha^n=1$ for some $n$. Is there any direct and simple argument to prove that if $\alpha_1^{m_1}=\alpha_2^{m_2}=\cdots=\alpha_k^{m_k}=1$ then $\alpha_1+\cdots + \alpha_k$ is an algebraic integer? (We can assume that $m_1=m_2=\cdots=m_k$). In other words, is there way to avoid the very general statement in (1)?