Levi-Civita field

In mathematics, the Levi-Civita field, named after Tullio Levi-Civita,^[1] is a non-Archimedean ordered field; i.e., a system of numbers containing infinite and infinitesimal quantities. It is usually denoted ${\displaystyle {\mathcal {R}}}$.

Each member ${\displaystyle a}$ can be constructed as a formal series of the form

${\displaystyle a=\sum _{q\in \mathbb {Q} }a_{q}\varepsilon ^{q},}$

where ${\displaystyle \mathbb {Q} }$ is the set of rational numbers, the coefficients ${\displaystyle a_{q}}$ are real numbers, and ${\displaystyle \varepsilon }$ is to be interpreted as a fixed positive infinitesimal. We require that for every rational number ${\displaystyle r}$, there are only finitely many ${\displaystyle q\in \mathbb {Q} }$ less than ${\displaystyle r}$ with ${\displaystyle a_{q}\neq 0}$; this restriction is necessary in order to make multiplication and division well defined and unique. Two such series are considered equal only if all their coefficients are equal. The ordering is defined according to the dictionary ordering of the list of coefficients, which is equivalent to the assumption that ${\displaystyle \varepsilon }$ is an infinitesimal.

The real numbers are embedded in this field as series in which all of the coefficients vanish except ${\displaystyle a_{0}}$.