
In
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Math ...
, a
number field
In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension).
Thus K is a ...
''F'' is called totally real if for each
embedding
In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup.
When some object X is said to be embedded in another object Y, the embedding is giv ...
of ''F'' into the
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
s the
image
An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensio ...
lies inside the
real number
In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
s. Equivalent conditions are that ''F'' is generated over Q by one
root
In vascular plants, the roots are the organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They are most often below the sur ...
of an
integer polynomial ''P'', all of the roots of ''P'' being real; or that the
tensor product algebra of ''F'' with the real field, over Q, is
isomorphic to a tensor power of R.
For example,
quadratic field
In algebraic number theory, a quadratic field is an algebraic number field of degree two over \mathbf, the rational numbers.
Every such quadratic field is some \mathbf(\sqrt) where d is a (uniquely defined) square-free integer different from 0 ...
s ''F'' of degree 2 over Q are either real (and then totally real), or complex, depending on whether the
square root
In mathematics, a square root of a number is a number such that ; in other words, a number whose ''square'' (the result of multiplying the number by itself, or ⋅ ) is . For example, 4 and −4 are square roots of 16, because .
...
of a positive or negative number is adjoined to Q. In the case of
cubic field In mathematics, specifically the area of algebraic number theory, a cubic field is an algebraic number field of degree three.
Definition
If ''K'' is a field extension of the rational numbers Q of degree 'K'':Qnbsp;= 3, then ''K'' is called ...
s, a cubic integer polynomial ''P''
irreducible over Q will have at least one real root. If it has one real and two complex roots the corresponding cubic extension of Q defined by adjoining the real root will ''not'' be totally real, although it is a
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
of real numbers.
The totally real number fields play a significant special role in
algebraic number theory. An
abelian extension In abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian. When the Galois group is also cyclic, the extension is also called a cyclic extension. Going in the other direction, a Galois extension is called solvabl ...
of Q is either totally real, or contains a totally real
subfield
Subfield may refer to:
* an area of research and study within an academic discipline
* Field extension, used in field theory (mathematics)
* a Division (heraldry)
* a division in MARC standards
MARC (machine-readable cataloging) standards ...
over which it has degree two.
Any number field that is
Galois over the
rationals
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ratio ...
must be either totally real or
totally imaginary.
See also
*
Totally imaginary number field In algebraic number theory, a number field is called totally imaginary (or totally complex) if it cannot be embedded in the real numbers. Specific examples include imaginary quadratic fields, cyclotomic field
In number theory, a cyclotomic field ...
*
CM-field In mathematics, a CM-field is a particular type of number field, so named for a close connection to the theory of complex multiplication. Another name used is J-field.
The abbreviation "CM" was introduced by .
Formal definition
A number field ' ...
, a totally imaginary quadratic extension of a totally real field
References
*{{Citation
, last=Hida
, first=Haruzo
, author-link=Haruzo Hida
, title=Elementary theory of L-functions and Eisenstein series
, year=1993
, publisher=
Cambridge University Press
Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press in the world. It is also the King's Printer.
Cambr ...
, series=London Mathematical Society Student Texts
, volume=26
, isbn=978-0-521-43569-7
Field (mathematics)
Algebraic number theory