We create the number field and the unique -adic valuation on . We normalize such that .
We create the extension by adjoining a rd root of and the unique extension of .
We check that has ramification jump , which means that .
We create the extension by adjoining a rd root of and the unique extension of .
We check that has ramification jump , which means that .
We create the extension and the unique extension of .
We check that the extension has ramification jump , which means that .
We check that the extension has ramification jumps and . This means that (in addition to the calculation in the previous cell) . To check this, we compute the Newton polygon of , where is the minimal polynomial of over .
We define the element . It has valuation .
We compute the minimal polynomial of over .
We compute the slopes of the Newton polygon of , which should give the two values and .
So, indeed, the roots of are equidistant. We now find the inductive description of the valuation corresponding to the smallest closed discoid containing all the roots of . In particular, .
As expected, is of the form .
We check that the minimal polynomial of cannot take the role of .