I have read that for a characteristic polynomial of a matrix $M$, say $$\sum_i \alpha_i T^{n-i},$$
where $\alpha_i=1$, satisfies $$\max_{i} |\alpha_i|^{1/i} \leq C |\det M|^{1/n} $$
for a certain constant $C$. Why is that true? Even for $n=2$ I am not at ease, it would be a bound of $a+b$ (the eigenvalues) by $\sqrt{ab}$?