Question on complete discrete valuation field.

Question

Let $F$ be a complete discrete valuation field and $f(X) = X^n + a_{n-1}X^{n-1} +\cdots+ a_0$ is an irreducible polynomial over $F$. How to show that

a) $ v(a_0) > 0$ implies $v(a_i) > 0$ for $i=1,...,n-1$;

b) if $v(a_0) \leq 0$, then $v(a_0) = \min(v(a_i)\mid i=0,...,n-1)$.

Edit. This is Exercise 2 on page 38 in Fesenko and Vostokov, Local Fields and Their Extensions.

Answer 1

Hints:

• If $L\mid F$ is a splitting field of $f$, then there is a unique extension $w \mid v$ of $v$ to $L$.
• If $\alpha$ is a root of $f$ in $L$, then $w(\alpha) = w(\sigma(\alpha))$.
• write $a_i$ as a symmetric polynomial in the roots $\alpha_i$ of $f$ and use the triangle inequality to get estimates on $v(\alpha_i)=w(\alpha_i)$.