Witt vector

Schmid[4] generalized further to non-commutative cyclic algebras of degree pn. In the process of doing so, certain polynomials related to addition of appeared. Witt seized on these polynomials. By using them systematically, he was able to give simple and unified constructions of degree pn field extensions and cyclic algebras. Specifically, he introduced a ring now called Wn(k), the ring of n-truncated p-typical Witt vectors. This ring has k as a quotient, and it comes with an operator F which is called the Frobenius operator because it reduces to the Frobenius operator on k. Witt observes that the degree pn analog of Artin–Schreier polynomials is

Representing elements in Fp as elements in the ring of Witt vectors W(Fp)Representing elements in Zp as elements in the ring of Witt vectors W(Fp)Additional properties of elements in the ring of Witt vectors motivating general definition

At this step, it becomes clear that one is actually working with addition of the form

The first few polynomials giving the sum and product of Witt vectors can be written down explicitly. For example,

The Witt polynomials for different primes p are special cases of universal Witt polynomials, which can be used to form a universal Witt ring (not depending on a choice of prime p). Define the universal Witt polynomials Wn for n ≥ 1 by

We can use these polynomials to define the ring of universal Witt vectors over any commutative ring R in much the same way as above (so the universal Witt polynomials are all homomorphisms to the ring R).

Similarly, the rings of truncated Witt vectors, and the rings of universal Witt vectors correspond to ring schemes, called the truncated Witt schemes and the universal Witt scheme.