Witt vectors

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 p is the ring of p-adic integers.

Any ${\displaystyle p}$-adic integer (an element of ${\displaystyle \mathbb {Z} _{p}}$, not to be confused with ${\displaystyle \mathbb {Z} /p\mathbb {Z} =\mathbb {F} _{p}}$) can be written as a power series ${\displaystyle a_{0}+a_{1}p^{1}+a_{2}p^{2}+\cdots }$, where the ${\displaystyle a}$'s are usually taken from the set ${\displaystyle \{0,1,2,\cdots ,p-1\}=\mathbb {F} _{p}}$. 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 ${\displaystyle a_{i}}$ from ${\displaystyle \mathbb {F} _{p}}$) is not the only possible choice, and Teichmüller suggested an alternative set of coefficients, taken from ${\displaystyle \mathbb {Z} _{p}}$ (that is, each ${\displaystyle a_{i}\in \mathbb {Z} _{p}}$) such that expressions for addition and multiplication can be written in closed form. These coefficients ${\displaystyle a_{i}}$ consist of 0 together with the ${\displaystyle p-1}$th roots of unity; that is, the roots of ${\displaystyle x^{p}-x=0}$ in ${\displaystyle \mathbb {Z} _{p},}$ so that ${\displaystyle a_{i}=a_{i}^{p}.}$

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