Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations...

algebraic number theory, a modulus (plural moduli) (or cycle, or extended ideal) is a formal product of places of a global field (i.e. an algebraic number...

study of algebraic number fields, that is, of algebraic extensions of the field of rational numbers, is the central topic of algebraic number theory. This...

complex numbers and techniques from analysis and calculus. Algebraic number theory employs algebraic structures such as fields and rings to analyze the properties...

algebraic number theory topics. These topics are basic to the field, either as prototypical examples, or as basic objects of study. Algebraic number field...

In algebraic number theory, a supersingular prime for a given elliptic curve is a prime number with a certain relationship to that curve. If the curve...

ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominent examples of commutative rings...

In abstract algebra, an adelic algebraic group is a semitopological group defined by an algebraic group G over a number field K, and the adele ring A...

no such polynomial exists then the number is called transcendental. More generally the theory deals with algebraic independence of numbers. A set of numbers...

the complex number 1 + i {\displaystyle 1+i} is algebraic because it is a root of x4 + 4. All integers and rational numbers are algebraic, as are all...

local field level. This technique is applied in algebraic number theory and modular representation theory. Hurwitz quaternion order – An example of ring...

publishes original research articles in algebra and number theory, interpreted broadly, including algebraic geometry and arithmetic geometry, for example....

In algebraic number theory, an algebraic integer is a complex number that is integral over the integers. That is, an algebraic integer is a complex root...

Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometrical problems...

algebra system SageMath Number Theory Library PARI/GP Fast Library for Number Theory Michael E. Pohst (1993): Computational Algebraic Number Theory,...

are much better understood than noncommutative ones. Algebraic geometry and algebraic number theory, which provide many natural examples of commutative...

theorem of Galois theory. The use of base fields other than Q is crucial in many areas of mathematics. For example, in algebraic number theory, one often does...

In algebraic number theory, a fundamental unit is a generator (modulo the roots of unity) for the unit group of the ring of integers of a number field...

having a proper algebraic covering) simple algebraic group defined over a number field is 1. Weil (1959) calculated the Tamagawa number in many cases of...

roots of such equations are called algebraic numbers – they are a principal object of study in algebraic number theory. Compared to Q ¯ {\displaystyle {\overline...

development of the theories of automorphic forms, representation theory of algebraic groups, and more advanced topics in algebraic number theory. The style is...

Algebraic K-theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. Geometric, algebraic, and arithmetic...

holes in them. Number theory is concerned with the properties of and relations between integers. Algebraic number theory applies algebraic methods and principles...

is called an algebraic matroid. No good characterization of algebraic matroids is known, but certain matroids are known to be non-algebraic; the smallest...

integers of an algebraic number field K {\displaystyle K} is the ring of all algebraic integers contained in K {\displaystyle K} . An algebraic integer is...

algebraic function fields, algebraic number fields, and p-adic fields are commonly used and studied in mathematics, particularly in number theory and...

example. In algebraic geometry over any field, by analogy, it also happens in algebraic codimension one. Ramification in algebraic number theory means a prime...

Williams: On the infrastructure of the principal ideal class of an algebraic number field of unit rank one. Math. Comp. 50 (1988), no. 182, 569–579. MR929554...

In algebraic number theory, a quadratic field is an algebraic number field of degree two over Q {\displaystyle \mathbf {Q} } , the rational numbers. Every...

elliptic curves (number theory) Hilbert's Nullstellensatz (theorem of zeroes) (commutative algebra, algebraic geometry) Hironaka theorem (algebraic geometry)...