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Complex number that solves a monic polynomial with integer coefficients
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_integer
Number in {..., –2, –1, 0, 1, 2, ...}
numbers. In algebraic number theory, integers are sometimes called rational integers to distinguish them from the more general algebraic integers. In fact
Integer
Type of complex number
{\displaystyle 1+i} is algebraic because it is a root of the polynomial x 4 + 4 {\displaystyle x^{4}+4} . Algebraic numbers include all integers, rational numbers
Algebraic_number
Algebraic construction
the ring of all algebraic integers contained in K {\displaystyle K} . An algebraic integer is a root of a monic polynomial with integer coefficients: x
Ring_of_integers
Finite extension of the rationals
any algebraic integer that is also a rational number must actually be an integer, hence the name "algebraic integer". Again using abstract algebra, specifically
Algebraic_number_field
Complex number whose mapping on a coordinate plane produces a triangular lattice
Eisenstein integers are a countably infinite set. The Eisenstein integers form a commutative ring Z[ω] of algebraic integers in the algebraic number field
Eisenstein_integer
Root of a quadratic polynomial with a unit leading coefficient
two whose coefficients are integers, i.e. quadratic integers are algebraic integers of degree two. Thus quadratic integers are those complex numbers that
Quadratic_integer
Branch of algebra that studies commutative rings
rings; rings of algebraic integers, including the ordinary integers Z {\displaystyle \mathbb {Z} } ; and p-adic integers. Commutative algebra is the main
Commutative_algebra
Complex number whose real and imaginary parts are both integers
properties. However, Gaussian integers do not have a total order that respects arithmetic. Gaussian integers are algebraic integers and form the simplest ring
Gaussian_integer
Algebraic structure with addition and multiplication
influenced by problems and ideas of algebraic number theory and algebraic geometry. In turn, commutative algebra is a fundamental tool in these branches
Ring_(mathematics)
Integers have unique prime factorizations
polynomial rings over a field. However, the theorem does not hold for algebraic integers. This failure of unique factorization is one of the reasons for the
Fundamental theorem of arithmetic
Fundamental_theorem_of_arithmetic
Branch of number theory
expressed in terms of properties of algebraic objects such as algebraic number fields and their rings of integers, finite fields, and function fields
Algebraic_number_theory
subfield of the field of algebraic numbers, and include the quadratic surds. Algebraic integer: A root of a monic polynomial with integer coefficients. Transfinite
List_of_types_of_numbers
Mathematical expression using basic operations
Abstract algebra. If the constants are restricted to integers, the set of numbers that can be described with an algebraic expression are called Algebraic numbers
Algebraic_expression
Theorem in transcendental number theory
α1, ..., αn are distinct algebraic numbers, then the exponentials eα1, ..., eαn are linearly independent over the algebraic numbers. This equivalence
Lindemann–Weierstrass_theorem
Branch of mathematics
between integers. Algebraic number theory applies algebraic methods and principles to this field of inquiry. Examples are the use of algebraic expressions
Algebra
Branch of pure mathematics
rational numbers), or defined as generalizations of the integers (for example, algebraic integers). Integers can be considered either in themselves or as solutions
Number_theory
Algebraic integer which represents an ideal in a ring of integers
In number theory, an ideal number is an algebraic integer which represents an ideal in the ring of integers of a number field; the idea was developed
Ideal_number
Mathematical structure in abstract algebra
is a *-algebra over R (where * is trivial). As a partial case, any *-ring is a *-algebra over integers. Any commutative *-ring is a *-algebra over itself
*-algebra
(Mathematical) decomposition into a product
such as certain rings of algebraic integers, which are not unique factorization domains. However, rings of algebraic integers satisfy the weaker property
Factorization
In mathematics, a non-algebraic number
is a real or complex number that is not algebraic: that is, not the root of a non-zero polynomial with integer (or, equivalently, rational) coefficients
Transcendental_number
Proposed lower bound on the Mahler measure for polynomials with integer coefficients
If P {\displaystyle P} has integer coefficients, this shows that M ( P ) {\displaystyle {\mathcal {M}}(P)} is an algebraic number so m ( P ) {\displaystyle
Lehmer's_conjecture
Type of programming language
sets, indices, algebraic expressions, powerful sparse index and data handling variables, constraints with arbitrary names. The algebraic formulation of
Algebraic_modeling_language
Branch of mathematics
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are sets with specific operations
Abstract_algebra
Generalization of algebraic integers
Hurwitz integer) is a quaternion whose components are either all integers or all half-integers (halves of odd integers; a mixture of integers and half-integers
Hurwitz_quaternion
The integer d {\displaystyle d} is called the radicand of the algebraic integer α {\displaystyle \alpha } . The norm of the quadratic algebraic number
Gauss_composition_law
Algebraic structure
the integers. Polynomial rings occur and are often fundamental in many parts of mathematics such as number theory, commutative algebra, and algebraic geometry
Polynomial_ring
the algebraic integers A {\displaystyle \mathbf {A} } with the ideal generated by p. Because p − 1 {\displaystyle p^{-1}} is not an algebraic integer, 1
Proofs of quadratic reciprocity
Proofs_of_quadratic_reciprocity
Type of algebraic integer
Pisot–Vijayaraghavan number (or Pisot number or PV number) is a real algebraic integer greater than 1, all of whose Galois conjugates are less than 1 in
Pisot–Vijayaraghavan_number
On algebraic independence of logarithms
combinations of logarithms of algebraic numbers. Nearly fifteen years earlier, Alexander Gelfond had considered the problem with only integer coefficients to be
Baker's_theorem
Roots of an algebraic element's minimal polynomial
mathematics, in particular field theory, the conjugate elements or algebraic conjugates of an algebraic element α, over a field extension L/K, are the roots of the
Conjugate element (field theory)
Conjugate_element_(field_theory)
Concept in algebraic number theory
{Q} ({\sqrt {-d}})} has class number 1. Equivalently, the ring of algebraic integers of Q ( − d ) {\displaystyle \mathbb {Q} ({\sqrt {-d}})} has unique
Heegner_number
Mathematics, group theory
{\chi _{i}(1)}{q}}\chi _{i}(g)} is an algebraic integer (since it is a sum of integer multiples of algebraic integers), which is absurd. This proves the
Burnside's_theorem
Describes statistically the splitting of primes in a given Galois extension of Q
numbers. Generally speaking, a prime integer will factor into several ideal primes in the ring of algebraic integers of K {\displaystyle K} . There are
Chebotarev_density_theorem
Matrix whose entries are integers
integer coefficients. Since the eigenvalues of a matrix are the roots of this polynomial, the eigenvalues of an integer matrix are algebraic integers
Integer_matrix
Algebra with unique prime factorization
insight into integer solutions of polynomial equations using rings of algebraic numbers of higher degree. For instance, fix a positive integer m {\displaystyle
Dedekind_domain
Application of geometry in number theory
number theory which uses geometry for the study of algebraic numbers. Typically, a ring of algebraic integers is viewed as a lattice in R n , {\displaystyle
Geometry_of_numbers
Number used for counting
2, 3, and so on, possibly excluding 0. The terms positive integers, non-negative integers, whole numbers, and counting numbers are also used. The set
Natural_number
Data type defined by combining other types
and type theory, an algebraic data type (ADT) is a composite data type, i.e. a type formed by combining other types. An algebraic data type is defined
Algebraic_data_type
Used to count, measure, and label
systems now called algebraic structures, which share certain properties of numbers, and may be seen as extending the concept. Some algebraic structures are
Number
In number theory, measure of non-unique factorization
{\displaystyle R} is a ring of algebraic integers, then the class number is always finite. This is one of the main results of classical algebraic number theory. Computation
Ideal_class_group
Polynomial with 1 as leading coefficient
that is integral over the integers is called an algebraic integer. This terminology is motivated by the fact that the integers are exactly the rational
Monic_polynomial
Measures the size of the ring of integers of the algebraic number field
discriminant of an algebraic number field is a numerical invariant that, loosely speaking, measures the size of the (ring of integers of the) algebraic number field
Discriminant of an algebraic number field
Discriminant_of_an_algebraic_number_field
Type of algebraic integer
In mathematics, a Salem number is a real algebraic integer α > 1 {\displaystyle \alpha >1} whose conjugate roots all have absolute value no greater than
Salem_number
Elements taken to zero by a homomorphism
a function that preserves the underlying algebraic structure in the domain to its image. When the algebraic structures involved have an underlying group
Kernel_(algebra)
Polynomial equation whose integer solutions are sought
involve finding integers that solve all equations simultaneously. Because such systems of equations define algebraic curves, algebraic surfaces, or, more
Diophantine_equation
Gives the rank of the group of units in the ring of algebraic integers of a number field
in algebraic number theory due to Peter Gustav Lejeune Dirichlet. It determines the rank of the group of units in the ring OK of algebraic integers of
Dirichlet's_unit_theorem
Rational numbers with root 5 added
1103/physrevb.35.5487. Rotman, Joseph J. (2017) [2002]. "Algebraic Integers". Advanced Modern Algebra. Vol. 2. American Mathematical Society. § C-5.3.2, pp
Golden_field
Algorithm for computing greatest common divisors
Algebra (2nd ed.). Menlo Park, CA: Addison–Wesley. pp. 190–194. ISBN 0-201-05487-6. Weinberger, P. (1973). "On Euclidean rings of algebraic integers"
Euclidean_algorithm
Arithmetic operation
the units (for example, 1 and −1 in the ring of integers). Another generalization of division to algebraic structures is the quotient group, in which the
Division_(mathematics)
Sum type in number theory
an algebraic integer in the pth cyclotomic field Q ( ζ p ) {\displaystyle \mathbb {Q} (\zeta _{p})} . The evaluation of the Gauss sum for an integer a
Quadratic_Gauss_sum
Integral domain in which the sum of two principal ideals is again a principal ideal
the algebraic integers there are no irreducible elements at all, since for any algebraic integer its square root (for instance) is also an algebraic integer
Bézout_domain
Algebraic structure with addition, multiplication, and division
Many other fields, such as fields of rational functions, algebraic function fields, algebraic number fields, finite fields, and p-adic fields are commonly
Field_(mathematics)
Algebraic study of differential equations
Systems Of Algebraic Differential Equations" and two books, Differential Equations From The Algebraic Standpoint and Differential Algebra. Ellis Kolchin
Differential_algebra
Modular function in mathematics
define the algebraic conjugates j(τ′) of j(τ) over Q(τ). Ordered by inclusion, the unique maximal order in Q(τ) is the ring of algebraic integers of Q(τ)
J-invariant
Type of mathematical expression
used to construct polynomial rings and algebraic varieties, which are central concepts in algebra and algebraic geometry. The word polynomial joins two
Polynomial
Every finite abelian extension of Q is contained within some cyclotomic field
is contained within some cyclotomic field. In other words, every algebraic integer whose Galois group is abelian can be expressed as a sum of roots of
Kronecker–Weber_theorem
Form of written communication for math
most algebraic integers are not integers and integers are specific algebraic integers. So, an algebraic integer is not an integer that is algebraic. Use
Language_of_mathematics
Commutative ring with a Euclidean division
(2003). Abstract Algebra (3 ed.). Wiley. p. 277. ISBN 978-0-471-43334-7. Weinberger, Peter J. (1973). "On Euclidean rings of algebraic integers". In Diamond
Euclidean_domain
Number-theoretic concept
In mathematics, a profinite integer is an element of the ring (sometimes pronounced as zee-hat or zed-hat) Z ^ = lim ← Z / n Z , {\displaystyle {\widehat
Profinite_integer
Polynomial with integer value for integer input
) Integer-valued polynomials are objects of study in their own right in algebra, and frequently appear in algebraic topology. The class of integer-valued
Integer-valued_polynomial
Ring that is also a vector space or a module
noncommutative algebraic geometry and, more recently, of derived algebraic geometry. See also: Generic matrix ring. A homomorphism between two R-algebras is an
Associative_algebra
Polynomial equation, generally univariate
example, x 5 − 3 x + 1 = 0 {\displaystyle x^{5}-3x+1=0} is an algebraic equation with integer coefficients and y 4 + x y 2 − x 3 3 + x y 2 + y 2 + 1 7 =
Algebraic_equation
Vector space equipped with a bilinear product
mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure
Algebra_over_a_field
Mathematical element
called algebraic integers. The algebraic integers in a finite extension field k of the rationals Q form a subring of k, called the ring of integers of k
Integral_element
3rd-century Greek mathematician
In modern use, Diophantine equations are algebraic equations with integer coefficients for which integer solutions are sought. Diophantine geometry
Diophantus
Algebraic manipulation of "true" and "false"
connection between his algebra and logic was later put on firm ground in the setting of algebraic logic, which also studies the algebraic systems of many other
Boolean_algebra
Decomposition of a number into a product
decomposition of a positive integer into a product of integers. Every positive integer greater than 1 is either the product of two or more integer factors greater
Integer_factorization
of integers is an order in the rational numbers (the only one). In an algebraic number field K {\displaystyle K} , an order is a ring of algebraic integers
Order_(ring_theory)
Subject area in mathematics
Algebraic K-theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. Geometric, algebraic, and arithmetic
Algebraic_K-theory
Mathematical ring with well-behaved ideals
mathematics are Noetherian (in particular the ring of integers, polynomial rings, and rings of algebraic integers in number fields), and many general theorems
Noetherian_ring
Branch of algebraic geometry
abstract development of algebraic geometry. Over finite fields, étale cohomology provides topological invariants associated to algebraic varieties. p-adic Hodge
Arithmetic_geometry
Commutative ring with no zero divisors other than zero
form an affine algebraic set that is not irreducible (that is, not an algebraic variety) in general. The only case where this algebraic set may be irreducible
Integral_domain
Mathematical terminology
classical algebraic number theory, let L be a Galois extension of a field K, and let G be the corresponding Galois group. Then the ring OL of algebraic integers
Galois_representation
Concept in field theory mathematics
positive integer. When I is a principal ideal αOK then N(I) is equal to the absolute value of the norm to Q of α, for α an algebraic integer. Field trace
Field_norm
Commutative algebra studies commutative rings, their ideals, and modules over such rings
rings; rings of algebraic integers, including the ordinary integers Z {\displaystyle \mathbb {Z} } ; and p-adic integers. Commutative algebra is the main
List of commutative algebra topics
List_of_commutative_algebra_topics
Concept in mathematical group theory
[G:C_{G}(x)]{\frac {\chi (x)}{\chi (1)}}} is an algebraic integer for all x in G. If F is algebraically closed and char(F) does not divide the order of
Character_theory
Number that is not a ratio of integers
real root of a polynomial with integer coefficients. Those that are not algebraic are transcendental. The real algebraic numbers are the real solutions
Irrational_number
Function in algebra
In algebra (in particular in algebraic geometry or algebraic number theory), a valuation is a function on a field that provides a measure of the size
Valuation_(algebra)
Algebraic structure used in analysis
in algebraic terms. The definition of a Lie algebra over a field extends to define a Lie algebra over any commutative ring R. Namely, a Lie algebra g {\displaystyle
Lie_algebra
Number system extending the rational numbers
integer (possibly negative), and each a i {\displaystyle a_{i}} is an integer such that 0 ≤ a i < p . {\displaystyle 0\leq a_{i}<p.} A p-adic integer
P-adic_number
Computation modulo a fixed integer
mathematics, modular arithmetic is a system of arithmetic operations for integers, differing from the usual ones in that numbers "wrap around" when reaching
Modular_arithmetic
Algebraic structure
{\displaystyle \langle x_{1},x_{2}\rangle } is not principal. Most rings of algebraic integers are not principal ideal domains. This is one of the main motivations
Principal_ideal_domain
Mathematical optimization problem restricted to integers
An integer programming, also known as integer optimization, problem is a mathematical optimization or feasibility program in which some or all of the variables
Integer_programming
Number with an integer power equal to 1
of the field is zero, the roots are complex numbers that are also algebraic integers. For fields with a positive characteristic, the roots belong to a
Root_of_unity
Product of a number by itself
squaring is quadratic. The square of an integer may also be called a square number or a perfect square. In algebra, the operation of squaring is often generalized
Square_(algebra)
Matrices named after Élie Cartan
\over (r_{j},r_{j})}} (sometimes called the Cartan integers) where ri are the simple roots of the algebra. The entries are integral from one of the properties
Cartan_matrix
Scientific area at the interface between computer science and mathematics
In mathematics and computer science, computer algebra, also called symbolic computation or algebraic computation, is a scientific area that refers to the
Computer_algebra
a sequence of natural numbers q0, q1, q2,... such that qnck is an algebraic integer in K for k = 0, 1, 2,..., n, and n = 0, 1, 2,... and for which q n
E-function
248-dimensional exceptional simple Lie group
means of a Chevalley basis for the Lie algebra, one can define E8 as a linear algebraic group over the integers and, consequently, over any commutative
E8_(mathematics)
Branch of algebra
commutative algebra, a major area of modern mathematics. Because these three fields (algebraic geometry, algebraic number theory and commutative algebra) are
Ring_theory
Ideal in a ring which has properties similar to prime elements
In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the
Prime_ideal
Type of algebraic number
In mathematics, a Perron number is an algebraic integer α which is real and greater than 1, but such that its conjugate elements are all less than α in
Perron_number
Condition under which an odd prime is a sum of two squares
in rings of quadratic integers. In summary, if O d {\displaystyle {\mathcal {O}}_{\sqrt {d}}} is the ring of algebraic integers in the quadratic field
Fermat's theorem on sums of two squares
Fermat's_theorem_on_sums_of_two_squares
Generalization of algebraic variety
In mathematics, specifically algebraic geometry, a scheme is a structure that enlarges the notion of an algebraic variety in several ways, such as taking
Scheme_(mathematics)
Algebraic structure
by the set of positive integers partially ordered by divisibility. An algebraic closure of a field serves also as an algebraic closure of any finite subextension
Finite_field
Number whose square is a given number
root of a positive integer, it is usually the positive square root that is meant. The square roots of an integer are algebraic integers—more specifically
Square_root
Mathematical software
algebraic decomposition Quantifier elimination over real numbers via cylindrical algebraic decomposition Mathematics portal List of computer algebra systems
Computer_algebra_system
About products of primitive polynomials
In algebra, Gauss's lemma, named after Carl Friedrich Gauss, is a theorem about polynomials over the integers, or, more generally, over a unique factorization
Gauss's_lemma_(polynomials)
Number, approximately 2.41421
{\sqrt {2}}-1} of the algebraic conjugate is smaller than 1, thus powers of σ {\displaystyle \sigma } generate almost integers and the sequence σ n
Silver_ratio
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