# Witt vector

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An element of an algebraic construct, first proposed by E. Witt  in 1936 in the context of the description of unramified extensions of -adic number fields. Witt vectors were subsequently utilized in the study of algebraic varieties over a field of positive characteristic , in the theory of commutative algebraic groups , , and in the theory of formal groups . Let be an associative, commutative ring with unit element. Witt vectors with components in are infinite sequences , , which are added and multiplied in accordance with the following rules:    where , are polynomials in the variables , with integer coefficients, uniquely defined by the conditions  where are polynomials, and is a prime number. In particular,  The Witt vectors with the operations introduced above form a ring, called the ring of Witt vectors and denoted by . For any natural number there also exists a definition of the ring of truncated Witt vectors of length . The elements of this ring are finite tuples , , with the addition and multiplication operations described above. The canonical mappings    are homomorphisms. The rule (or ) defines a covariant functor from the category of commutative rings with unit element into the category of rings. This functor may be represented by the ring of polynomials (or ) on which the structure of a ring object has been defined. The spectrum (or ) is known as a Witt scheme (or a truncated Witt scheme) and is a ring scheme .

Each element defines a Witt vector called the Teichmüller representative of the element . If is a perfect field of characteristic , is a complete discrete valuation ring of zero characteristic with field of residues and maximal ideal . Each element can be uniquely represented as where . Conversely, each such ring with field of residues is canonically isomorphic to the ring . The Teichmüller representation makes it possible to construct a canonical multiplicative homomorphism , splitting the mapping If is the prime field of elements, is the ring of integral -adic numbers .