In the 19th century, Ernst Eduard Kummer studied cyclic extensions of fields as part of his work on Fermat's Last Theorem. This led to the subject now known as Kummer theory. Let k be a field containing a primitive nth root of unity. Kummer theory classifies degree n cyclic field extensions K of k. Such fields are in bijection with order n cyclic groups , where corresponds to .
But suppose that k has characteristic p. The problem of studying degree p extensions of k, or more generally degree pn extensions, may appear superficially similar to Kummer theory. However, in this situation, k cannot contain a primitive pth root of unity. If x is a pth root of unity in k, then it satisfies . Because raising to the pth power is the Frobenius homomorphism, this equation may be rewritten as , and therefore . Consequently Kummer theory is never applicable to extensions whose degree is divisible by the characteristic.
The case where the characteristic divides the degree is now called Artin-Schreier theory because the first progress was made by Artin and Schreier. Their initial motivation was the Artin-Schreier theorem, which characterizes the real closed fields as those whose absolute Galois group has order two. This inspired them to ask what other fields had finite absolute Galois groups. In the midst of proving that no other such fields exist, they proved that degree p extensions of a field k of characteristic p were the same as splitting fields of Artin-Schreier polynomials. These are by definition of the form . By repeating their construction, they described degree p2 extensions. Abraham Adrian Albert used this idea to describe degree pn extensions. Each repetition entailed complicated algebraic conditions to ensure that the field extension was normal.
Schmid generalized further to non-commutative cyclic algebras of degree pn. In the process of doing so, certain polynomials related to addition of p-adic integers 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
where . To complete the analogy with Kummer theory, define to be the operator . Then the degree pn extensions of k are in bijective correspondence with cyclic subgroups of order pn, where corresponds to the field .
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 integer interval . It is hard to provide an algebraic expression for addition and multiplication using this representation, as one faces the problem of carrying between digits. However, taking representative coefficients is only one of many choices, and Hensel himself (the creator of p-adic numbers) suggested the roots of unity in the field as representatives. These representatives are therefore the number together with the roots of unity; that is, the solutions of in , so that . This choice extends naturally to ring extensions of in which the residue field is enlarged to with , some power of . Indeed, it is these fields (the fields of fractions of the rings) that motivated Hensel's choice. Now the representatives are the solutions in the field to . Call the field , with an appropriate primitive root of unity (over ). The representatives are then and for .
Since these representatives form a multiplicative set they can be thought of as characters. Some thirty years after Hensel's works Teichmüller studied these characters, which now bear his name, and this led him to a characterisation of the structure of the whole field in terms of the residue field. These Teichmüller representatives can be identified with the elements of the finite field of order by taking residues modulo in , and elements of are taken to their representatives by the Teichmüller character. This operation identifies the set of integers in with infinite sequences of elements of .
Taking those representatives the expressions for addition and multiplication can be written in closed form. We now have the following problem (stated for the simplest case: ): 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.
Detailed motivational sketch
We derive the ring of -adic integers from the finite field using a construction which naturally generalizes to the Witt vector construction.
The ring of -adic integers can be understood as the projective limit of Specifically, it consists of the sequences with such that for That is, each successive element of the sequence is equal to the previous elements modulo a lower power of p; this is the inverse limit of the projections .
where are usually taken from the integer interval 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. We will sketch of a method of defining ring operations for such power series.
Letting be denoted by , one might consider the following definition for addition:
and one could make a similar definition for multiplication. However, this is not a closed formula, since the new coefficients are not in the allowed set
There is a better coefficient subset of which does yield closed formulas, the Teichmuller representatives: zero together with the roots of unity. They can be explicitly calculated (in terms of the original coefficient representatives ) as roots of through Hensel lifting, the p-adic version of Newton's method. 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 sequence The existence of a lift in each step is guaranteed by the greatest common divisor in every
This algorithm shows that for every , there is exactly one Teichmuller representative with , which we denote Indeed, this defines the Teichmüller charactersatisfying if we denote Note that is not additive, as the sum need not be a representative. Despite this, if in then in
Because of this one-to-one correspondence given by , 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 . Hence, . The process is then repeated, subtracting and proceed likewise. This yields a sequence of congruences
Hence we have a power series for each residue of x modulo powers of p, but with coefficients in the Teichmüller representatives rather than . It is clear that
for all as so the difference tends to 0 with respect to the p-adic metric. The resulting coefficients will typically differ from the 's modulo , except the first one.
The Teichmuller coefficients have the key additional property that which is missing for the numbers in . This can be used to describe addition, as follows. Since the Teichmüller character is not additive, is not true in . But it holds in as the first congruence implies. In particular,
This completely determines by the lift. Moreover, the congruence modulo indicates that the calculation can actually be done in satisfying the basic aim of defining a simple additive structure.
For this step is already very cumbersome. Write
Just as for a single th power is not enough: one must take
However, is not in general divisible by but it is divisible when in which case combined with similar monomials in will make a multiple of .
At this step, it becomes clear that one is actually working with addition of the form
This motivates the definition of Witt vectors.
Construction of Witt rings
Fix a prime numberp. A Witt vector over a commutative ring R is a sequence: of elements of R. Define the Witt polynomials by
and in general
The are called the ghost components of the Witt vector , and are usually denoted by The ghost components can be thought of as an alternative coordinate system for the R-module of sequences.
The ring of Witt vectors is defined by componentwise addition and multiplication of the ghost components. That is, that there is a unique way to make the set of Witt vectors over any commutative ring R into a ring such that:
the sum and product are given by polynomials with integral coefficients that do not depend on R, and
projection to each ghost component is a ring homomorphism from the Witt vectors over R, to R.
In other words,
and are given by polynomials with integral coefficients that do not depend on R, and
The first few polynomials giving the sum and product of Witt vectors can be written down explicitly. For example,
These are to be understood as shortcuts for the actual formulas. If for example the ring R has characteristic p, the division by p in the first formula above, the one by that would appear in the next component and so forth, do not make sense. However, if the p-power of the sum is developed, the terms are cancelled with the previous ones and the remaining ones are simplified by p, no division by p remains and the formula makes sense. The same consideration applies to the ensuing components.
The Witt ring of any commutative ring R in which p is invertible is just isomorphic to (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 , 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 written in terms of the Teichmuller representatives, as 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.
Universal Witt vectors
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 vector of 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).
Witt also provided another approach using generating functions.
Let be a Witt vector and define
For let denote the collection of subsets of whose elements add up to . Then
if are the respective coefficients in the power series . Then
Since is a polynomial in and likewise for , we can show by induction that is a polynomial in
If we set then
Now 3-tuples with are in bijection with 3-tuples with , via ( is the least common multiple), our series becomes
where 's are polynomials of So by similar induction, suppose
then can be solved as polynomials of
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 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 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.
Commutative unipotent algebraic groups
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.