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**GaloisGroup** A valuation is a function $\displaystyle v: K(field)\to \mathbb R $ so that

$\displaystyle

1) v(0) = \infty\\

2) v(xy) = v(x)+v(y)\\

3) v(x+y)\geq min\{v(x),v(y)\}

$

Consider the field $\displaystyle F(x)$ of rational functions in one variable over a field $\displaystyle F$.

Let $\displaystyle p\in F[x] $ be a monic irreducible polynomial.

Define the map $\displaystyle \nu_{p}: F(x)\to \mathbb Z\cup \{\infty\}$ as follows:

For a nonzero polynomial $\displaystyle f\in F[x]$ with $\displaystyle p^d||f$, that is,

$\displaystyle p^d|f $ and $\displaystyle p^{d+1}\nmid f,$ set $\displaystyle \nu_p(f) = d;$

For a nonzero rational function $\displaystyle f/g\in F(x)$, set $\displaystyle \nu_p(f/g)=\nu_p(f)-\nu_p(g);$

Set $\displaystyle \nu_p(0) =\infty.$

How can I see that this map $\displaystyle \nu_p $ is a valuation. My notes claims that its valuation ring is:

$\displaystyle \mathfrak o = \left\{ \frac{f}{g} : f,g\in F[x], \; p\nmid g\right\}, $

and its maximal ideal is $\displaystyle \mathfrak m = \left\{ \frac{f}{g} : f,g\in F[x], \; p|f, \; p\nmid g \right\}. $

How can I see that $\displaystyle \mathfrak o $ is its valuation ring and $\displaystyle \mathfrak m $ is its maximal ideal?