Any -adic integer (an element of , not to be confused with ) can be written as a power series, where the 's are usually taken from the set . However, it is hard to provide an algebraic expression for addition and multiplication, using this representation for the p-adic integers, as one faces the problem of carrying. However, this set of representative coefficients (that is, taking the coefficients from ) is not the only possible choice, and Teichmüller suggested an alternative set of coefficients, taken from (that is, each ) such that expressions for addition and multiplication can be written in closed form. These coefficients consist of 0 together with the th roots of unity; that is, the roots of in so that
These Teichmüller representatives can be identified with the elements of the finite field of order (by taking residues modulo ), and elements of are taken to their representatives by the Teichmüller character. This identifies the set of -adic integers with infinite sequences of elements of .
We now have the following problem: given two infinite sequences of elements of , describe their sum and product as -adic integers explicitly. This problem was solved by Witt using Witt vectors.
The below derives the ring of -adic integers from the finite field with elements, , using a construction which naturally generalizes to the Witt vector construction.
The ring of -adic integers can be understood as the projective limit of taking . Specifically, it consists of the sequences with , such that if That is, the next element of the sequence equals the last, modulo one more power of p; this gives the projection defining the inverse limit.
The elements of can be expanded as (formal) power series in , where the 's are usually taken from the set . Of course, this power series usually will not converge in using the standard metric on the reals, but it will converge in , with the p-adic metric. Viewed as a sequence, is just , if one forgets the ring structure. What follows is a sketch of how a ring structure can be provided for the sequence.
Letting be denoted by , one might consider the following definition for addition:
However, this lacks several properties needed to produce a general formula; most notably, one does not have that
There is an alternative subset of which can be used as the coefficient set. This is the set of Teichmüller representatives of elements of . Without they form a subgroup of , identified with through the Teichmüller character. Note that is not additive, as the sum need not be a representative. Despite this, if in , then in . This is conceptually justified by if we denote .
Teichmüller representatives are explicitly calculated as roots of through Hensel lifting. For example, in , to calculate the representative of , one starts by finding the unique solution of in with ; one gets . Repeat this in , with the conditions and gives , and so on; the resulting Teichmüller representative is The existence of a lift in each step is guaranteed by the greatest common divisor in every .
Note for every , there is exactly one representative, namely , with . Because of this one-to-one correspondence, one can expand every -adic integer as a power series in , with coefficients taken from the Teichmüller representatives. An explicit algorithm can be given, as follows. Write the Teichmüller representative as . Then, if one has some arbitrary p-adic integer of the form , one takes the difference , leaving a value divisible by . The process is then repeated, subtracting and proceed likewise. The resulting coefficients will typically differ from the 's modulo , except the first one.
This transformed sequence of coefficients has an additional property, namely that , which the original sequence did not have. This can be used to describe addition, as follows. Since the Teichmüller character is not additive, it is not true that in . But it does hold in , as the first congruence implies. In particular, , and thus
The Witt ring of any commutative ring R in which p is invertible is just isomorphic to RN (the product of a countable number of copies of R). In fact the Witt polynomials always give a homomorphism from the ring of Witt vectors to RN, and if p is invertible this homomorphism is an isomorphism.
The Witt ring of the finite field of order p is the ring of p-adic integers, as is demonstrated above.
The Witt ring of a finite field of order pn is the unramified extension of degree n of the ring of p-adic integers.
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
and in general
Again, is called the ghost components of the Witt vector , and is usually denoted 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).
The map taking a commutative ring R to the ring of Witt vectors over R (for a fixed prime p) is a functor from commutative rings to commutative rings, and is also representable, so it can be thought of as a ring scheme, called the Witt scheme, over Spec(Z). The Witt scheme can be canonically identified with the spectrum of the ring of symmetric functions.
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.
Moreover, the functor taking the commutative ring to the set is represented by the affine space, and the ring structure on Rn makes into a ring scheme denoted . From the construction of truncated Witt vectors, it follows that their associated ring scheme is the scheme with the unique ring structure such that the morphism given by the Witt polynomials is a morphism of ring schemes.
Over an algebraically closed field of characteristic 0, any unipotent abelian connected algebraic group is isomorphic to a product of copies of the additive group . The analogue of this for fields of characteristic p is false: the truncated Witt schemes are counterexamples. (We make them into algebraic groups by forgetting the multiplication and just using the additive structure.) However, these are essentially the only counterexamples: over an algebraically closed field of characteristic p, any unipotent abelian connected algebraic group is isogenous to a product of truncated Witt group schemes.