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.
can be constructed as a formal series of the form where
is the set of rational numbers, the coefficients
is to be interpreted as a fixed positive infinitesimal.
We require that for every rational number
; 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
The real numbers are embedded in this field as series in which all of the coefficients vanish except
are two Levi-Civita series, then (One can check that for every
is finite, so that all the products are well-defined, and that the resulting series defines a valid Levi-Civita series.)
Equipped with those operations and order, the Levi-Civita field is indeed an ordered field extension of
The Levi-Civita field is real-closed, meaning that it can be algebraically closed by adjoining an imaginary unit (i), or by letting the coefficients be complex.
It is rich enough to allow a significant amount of analysis to be done, but its elements can still be represented on a computer in the same sense that real numbers can be represented using floating point.
It is the basis of automatic differentiation, a way to perform differentiation in cases that are intractable by symbolic differentiation or finite-difference methods.
[2] The Levi-Civita field is also Cauchy complete, meaning that relativizing the
definitions of Cauchy sequence and convergent sequence to sequences of Levi-Civita series, each Cauchy sequence in the field converges.
Equivalently, it has no proper dense ordered field extension.
As an ordered field, it has a natural valuation given by the rational exponent corresponding to the first non zero coefficient of a Levi-Civita series.
The valuation ring is that of series bounded by real numbers, the residue field is
The resulting valued field is Henselian (being real closed with a convex valuation ring) but not spherically complete.
Indeed, the field of Hahn series with real coefficients and value group
is a proper immediate extension, containing series such as
of Puiseux series over the field of real numbers, that is, it is a dense extension of
without proper dense extension.
Here is a list of some of its notable proper subfields and its proper ordered field extensions: