Levi-Civita field

In mathematics, the Levi-Civita field, named after Tullio Levi-Civita, is a non-Archimedean ordered field; i.e., a system of numbers containing infinite and infinitesimal quantities. Each member formula_1 can be constructed as a formal series of the form
where formula_3 are real numbers, formula_4 is the set of rational numbers, and formula_5 is to be interpreted as a positive infinitesimal. The support of formula_1, i.e., the set of indices of the nonvanishing coefficients formula_7 must be a left-finite set: for any member of formula_4, there are only finitely many members of the set less than it; this restriction is necessary in order to make multiplication and division well defined and unique. The ordering is defined according to the dictionary ordering of the list of coefficients, which is equivalent to the assumption that formula_5 is an infinitesimal.
The real numbers are embedded in this field as series in which all of the coefficients vanish except formula_10.
Definition of the field operations and positive cone.
If formula_24 and formula_25 are two Levi-Civita series, then 
Equipped with those operations and order, the Levi-Civita field is indeed an ordered field extension of formula_36 where the series formula_5 is a positive infinitesimal.
Properties and applications.
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.
The Levi-Civita field is also Cauchy complete, meaning that relativizing the formula_38 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 formula_36, and the value group is formula_40. 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 formula_40 is a proper immediate extension, containing series such as formula_42 which are not in the Levi-Civita field.
Relations to other ordered fields.
The Levi-Civita field is the Cauchy-completion of the field formula_43 of Puiseux series over the field of real numbers, that is, it is a dense extension of formula_43 without proper dense extension. Here is a list of some of its notable proper subfields and its proper ordered field extensions: