Today we’re going to accomplish the original goal we set out for ourselves. We will construct the ring of generalized Witt vectors. First, let be a collection of indeterminates. We can define an infinite collection of polynomials in using the following formulas:
and in general .
Now let . This just an arbitrary two variable polynomial with coefficients in .
We can define new polynomials such that the following condition is met .
In short we’ll notate this . The first thing we need to do is make sure that such polynomials exist. Now it isn’t hard to check that the can be written as a -linear combination of the just by some linear algebra.
, and , etc. so we can plug these in to get the existence of such polynomials with coefficients in . It is a fairly tedious lemma to prove that the coefficients are actually in , so we won’t detract from the construction right now to prove it.
Define yet another set of polynomials , and by the following properties:
, and .
We now can construct , the ring of generalized Witt vectors over . Define to be the set of all infinite sequences with entries in . Then we define addition and multiplication by and .
Next time we’ll actually check that this is a ring for any . To show you that this isn’t as horrifyingly strange and arbitrary as it looks, it turns out that all these rules boil down to just the -adic integers when is the finite field of order , i.e. . It also turns out that there is a much cleaner construction of this than element-wise if all you care about are the existence and certain properties.