A valuation is a function so that

Consider the field of rational functions in one variable over a field .

Let be a monic irreducible polynomial.

Define the map as follows:

For a nonzero polynomial with , that is,

and set

For a nonzero rational function , set

Set

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

and its maximal ideal is

How can I see that is its valuation ring and is its maximal ideal?