In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a Witt vector is an
infinite sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called t ...
of elements of a
commutative ring
In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not ...
.
Ernst Witt showed how to put a ring
structure
A structure is an arrangement and organization of interrelated elements in a material object or system, or the object or system so organized. Material structures include man-made objects such as buildings and machines and natural objects such a ...
on the set of Witt vectors, in such a way that the ring of Witt vectors
over the finite field of order
is the ring of
-adic integers. They have a highly non-intuitive structure
upon first glance because their additive and multiplicative structure depends on an infinite set of recursive formulas which do not behave like addition and multiplication formulas for standard p-adic integers. The main idea
behind Witt vectors is instead of using the standard
-adic expansion
to represent an element in
, we can instead consider an expansion using the
Teichmüller characterwhich sends each element in the solution set of
in
to an element in the solution set of
in
. That is, we expand out elements in
in terms of roots of unity instead of as profinite elements in
. We can then express a
-adic integer as an infinite sum
which gives a Witt vector
Then, the non-trivial additive and multiplicative structure in Witt vectors comes from using this map to give
an additive and multiplicative structure such that
induces a commutative ring morphism.
History
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
be a field containing a primitive
-th root of unity. Kummer theory classifies degree
cyclic field extensions
of
. Such fields are in bijection with order
cyclic groups
, where
corresponds to
.
But suppose that
has characteristic
. The problem of studying degree
extensions of
, or more generally degree
extensions, may appear superficially similar to Kummer theory. However, in this situation,
cannot contain a primitive
-th root of unity. Indeed, if
is a
-th root of unity in
, then it satisfies
. But consider the expression
. By expanding using
binomial coefficients we see that the operation of raising to the
-th power, known here as the
Frobenius homomorphism
In commutative algebra and field theory, the Frobenius endomorphism (after Ferdinand Georg Frobenius) is a special endomorphism of commutative rings with prime characteristic , an important class which includes finite fields. The endomorphism m ...
, introduces the factor
to every coefficient except the first and the last, and so modulo
these equations are the same. 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 today 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 field
In mathematics, a real closed field is a field ''F'' that has the same first-order properties as the field of real numbers. Some examples are the field of real numbers, the field of real algebraic numbers, and the field of hyperreal numbers.
D ...
s 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
extensions of a field
of characteristic
were the same as splitting fields of ''Artin–Schreier polynomials''. These are by definition of the form
By repeating their construction, they described degree
extensions.
Abraham Adrian Albert used this idea to describe degree
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
. In the process of doing so, certain polynomials related to the addition of
-adic integers appeared. Witt seized on these polynomials. By using them systematically, he was able to give simple and unified constructions of degree
field extensions and cyclic algebras. Specifically, he introduced a ring now called
, the ring of
-truncated
-typical Witt vectors. This ring has
as a quotient, and it comes with an operator
which is called the Frobenius operator because it reduces to the Frobenius operator on
. Witt observes that the degree
analog of Artin–Schreier polynomials is
:
where
. To complete the analogy with Kummer theory, define
to be the operator
Then the degree
extensions of
are in bijective correspondence with cyclic subgroups
of order
, where
corresponds to the field
.
Motivation
Any
-adic integer (an element of
, not to be confused with
) can be written as a
power series
In mathematics, a power series (in one variable) is an infinite series of the form
\sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots
where ''an'' represents the coefficient of the ''n''th term and ''c'' is a con ...
, where the
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