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Concept
Witt vector
Summary
In mathematics, a Witt vector is an infinite sequence of elements of a commutative ring. Ernst Witt showed how to put a ring structure on the set of Witt vectors, in such a way that the ring of Witt vectors over the finite field of order is isomorphic to , 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 expansionto 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 sumwhich gives a Witt vectorThen, 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.
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, introduces the factor to every coefficient except the first and the last, and so modulo these equations are the same.
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