Algebraic number theory

thumb|upright=1.25|The integers arranged on a number line

additive subgroup of a ring closed under multiplcation by arbitrary ring element

In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the roots without computing them. More precisely, it is a polynomial function of the coefficients of the original polynomial. The discriminant is widely used in polynomial factoring, number theory, and algebraic geometry.

major branch of number theory

a finite degree (and hence algebraic) field extension of the field of rational numbers

function that can be defined as the root of a polynomial equation

theorem

integral domain where every nonzero element is uniquely expressible as a product of prime elements

algebraic field extension where the automorphism group fixes precisely the base field

in mathematics, an invertible element or a unit in a ring R

algebraic construction

algebraic number field of degree two over the rational numbers

Type of prime number

ring with unique factorization for ideals (mathematics)

field extension of the rational numbers by a primitive root of unity

endomorphism of a commutative ring of non-zero characteristics

square-free positive integer such that the corresponding imaginary quadratic field has unique factorization; one of 1, 2, 3, 7, 11, 19, 43, 67, or 163

A generalization of the Riemann zeta function for algebraic number fields

field extension whose Galois group is abelian

mathematical concept

non-discrete locally compact topological field

mathematical set in algebraic number theory

Concept in field theory mathematics

in numbers theory, a given integer a which is not a perfect square and not −1 is a primitive root modulo infinitely many primes p

generalization of the ring-theoretical notion of ideal to integral domains

commutative ring, whose elements (called adeles) are an infinite tuple of elements from each completion of a number field, such that a cofinite number of them lie in the ring of algebraic integers; "adele" is short for "additive ideal element"

mathematical hypothesis

type of valuation

mathematical theory describing field extensions involving the adjunction of nth roots

module over the group algebra of some Galois group

principle that an integer equation can be solved by piecing together modular solutions

mathematic formula

Group comohology of Galois modules

algebraic integer of a given quadratic field

abelian group related to division algebras

cohomology theory associated to a group 𝐺 and a 𝐺‐module

invariant that measures the size of the ring of integers of the algebraic number field

conditions under which the congruence x3 equals p (mod q) is solvable

mathematical problem

tool for solving algebraic equations with valued coefficients

branching out of a mathematical structure

special point on an algebraic curve

area of mathematics that is a combination of algebraic number theory and topology

a number field K such that, for each embedding of K into the complex numbers, the image lies inside the real numbers

the ideal-theoretic generalization of the field norm

polynomial that preserves addition

mathematical concept

semitopological group in abstract algebra