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Jun
19
comment Characteristic polynomial and $p$-adic valuation
Let $r_{i}$ be the roots of $f(x)$. Then by the relations between roots and coefficients, I know that $|r_{1} + \cdots + r_{n}|_{p} = |a_{1}|_{p}$. As the $p$-adic valuation is nonarchimedean, we know that $|a_{1}|_{p} \leq \max(|r_{1}|_{p}, \ldots, |r_{n}|_{p})$. How can I prove that $|a_{1}|_{p} \leq 1/p$?
Jun
19
asked Characteristic polynomial and $p$-adic valuation