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,
For a nonzero rational function , 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?