the ring of Witt vectors of length . Its elements are called Witt vectors of lenth .
may be considered as a set with an unusual ring structure.
Let us first fix a positive integer and examine the kernel of a map
where is the natural projection. Since
In other words, for some integer . On the other hand, we have
This implies that and therefore we have an inclusion
which turns to be a bijection ( ).
We then take a projective limit of the both hand sides and obtain the resired isomorphism.