In number theory, a number field F is called totally real if for each embedding of F into the complex numbers the image lies inside the real numbers....

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...

In mathematics, a real number is a number that can be used to measure a continuous one-dimensional quantity such as a distance, duration or temperature...

retrieved 2004-10-13 Woodin, W. H.; Dales, H. G. (1996), Super-real fields: totally ordered fields with additional structure, Oxford: Clarendon Press, ISBN 978-0-19-853991-9...

mathematics, a real closed field is a field F that has the same first-order properties as the field of real numbers. Some examples are the field of real numbers...

In mathematics, the surreal number system is a totally ordered proper class containing not only the real numbers but also infinite and infinitesimal numbers...

three real roots, then K is called a totally real cubic field and it is an example of a totally real field. If, on the other hand, f has a non-real root...

π: Every point of the number line corresponds to a unique real number, and every real number to a unique point. Using a number line, numerical concepts...

rational number is a real number. The real numbers that are rational are those whose decimal expansion either terminates after a finite number of digits...

In mathematics, the extended real number system is obtained from the real number system R {\displaystyle \mathbb {R} } by adding two infinity elements:...

extension of number fields with degree greater than 1 and the units groups for the integers of L and K have the same rank then K is totally real and L is...

originally stated that, assuming that F {\displaystyle F} is a totally real number field and that F ∞ / F {\displaystyle F_{\infty }/F} is the cyclotomic...

class number one. The next totally real cubic field (of discriminant 1957) has class number two. The polynomials defining the totally real cubic fields that...

In mathematics, an algebraic number field (or simply number field) is an extension field K {\displaystyle K} of the field of rational numbers Q {\displaystyle...

"Enumeration of totally real number fields of bounded root discriminant", in van der Poorten, Alfred J.; Stein, Andreas (eds.), Algorithmic number theory. Proceedings...

operations on rational and real numbers. A field is thus a fundamental algebraic structure which is widely used in algebra, number theory, and many other...

of Abelian varieties with multiplications from an order in a totally real number field). His doctoral students include Paul Monsky, Timothy J. Hickey...

is a real algebra and that can be seen to be totally ordered. The field of fractions F of A is a superreal field if F strictly contains the real numbers...

\zeta _{F}} . More specifically, let F be a totally real number field and let N be the largest natural number such that the extension of F by the Nth root...

numbers, effective numbers or the computable reals or recursive reals. The concept of a computable real number was introduced by Émile Borel in 1912, using...

superordered field is a totally real field in which the set of sums of squares forms a fan. Linearly ordered group – Group with translationally invariant total order;...

Taniyama 1961). A number field K is a CM-field if it is a quadratic extension K/F where the base field F is totally real but K is totally imaginary. I.e...

closure of a strict total order < {\displaystyle <} is a (non-strict) total order. Any subset of a totally ordered set X is totally ordered for the restriction...

fields isomorphic to R {\displaystyle \mathbb {R} } or C {\displaystyle \mathbb {C} } . The totally real number fields are those for which only real fields...

Kroneckerian field A totally real algebraic number field or a totally imaginary quadratic extension of a totally real field. CM-field or J-field An algebraic...

The axiom is crucial in the characterization of the reals. For example, the totally ordered field of the rational numbers Q satisfy the first three axioms...

each a complex number in the upper half-plane, satisfying a modular relation for 2×2 matrices with entries in a totally real number field. Siegel modular...

high-definition. The seventh season of Totally Spies! was announced by French television network Gulli in a Youtube video posted by Totally Spies! youtube channel on...

varieties are solutions have been likewise identified. Let F be a totally real number field and D a quaternion division algebra over F. The multiplicative...

be a totally real number field of degree m over the rational field. Let σ 1 , … , σ m {\displaystyle \sigma _{1},\ldots ,\sigma _{m}} be the real embeddings...