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...

their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic objects such as algebraic number fields and their rings...

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...

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...

a non-zero algebraic number. Then, since eiπ = −1 is algebraic (see Euler's identity), iπ must be transcendental. But since i is algebraic, π must therefore...

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

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...

is an algebraic number. Fields of algebraic numbers are also called algebraic number fields, or shortly number fields. Algebraic number theory studies...

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

Universal algebra is still more abstract in that it is not interested in specific algebraic structures but investigates the characteristics of algebraic structures...

are algebraic was first published by Georg Cantor in his 1874 paper "On a Property of the Collection of All Real Algebraic Numbers". Non-algebraic numbers...

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

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

algebraic numbers. Consider the approximation of a complex number x by algebraic numbers of degree ≤ n and height ≤ H. Let α be an algebraic number of...

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...

similarly. An irrational number may be algebraic, that is a real root of a polynomial with integer coefficients. Those that are not algebraic are transcendental...

In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are sets with specific operations...

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...

an algebraic function is a function that can be defined as the root of an irreducible polynomial equation. Algebraic functions are often algebraic expressions...

In algebraic number theory, the conductor of a finite abelian extension of local or global fields provides a quantitative measure of the ramification in...

In algebraic geometry, divisors are a generalization of codimension-1 subvarieties of algebraic varieties. Two different generalizations are in common...

science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, model theory, number theory...

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...

equations that involve nth roots and, more generally, algebraic expressions. This makes the term algebraic equation ambiguous outside the context of the old...

an important tool and object of study in commutative algebra, algebraic number theory and algebraic geometry. The prime ideals of the ring of integers are...

branches like algebraic number theory and algebraic topology. The word algebra itself has several meanings. Algebraic may also refer to: Algebraic data type...

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

and A has the structure of an associative algebra over R. Involutive algebras generalize the idea of a number system equipped with conjugation, for example...

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

{Q} } ⁠ are called algebraic number fields, and the algebraic closure of ⁠ Q {\displaystyle \mathbb {Q} } ⁠ is the field of algebraic numbers. In mathematical...