Mediaspace scheduled maintenance: Aug 25, 2026 07:00 - 12:00 AM. During this time, videos will be temporarily unavailable. Check status updates.

Lecture

Integers and Rings

In course

MATH-215: Algebra III - rings and fields

This is an introductory course in ring and field theory. ...

Description

This lecture covers the concepts of integers, rings, subrings, and the body of fractions. It delves into the properties of rings, including invertibility and divisors of zero. The lecture also discusses equivalence relations and the integration of formal fractions.

This video is available exclusively on Mediaspace for a restricted audience. Please log in to MediaSpace to access it if you have the necessary permissions.

Watch on Mediaspace

Instructor

Zsolt Patakfalvi

See CV here .

Official source

Ontological neighbourhood

Mathematics

Algebra: Abstract algebra

Related concepts (32)

Equivalence relation

In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relation provides a partition of the underlying set into disjoint equivalence classes. Two elements of the given set are equivalent to each other if and only if they belong to the same equivalence class.

Equivalence class

In mathematics, when the elements of some set have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set into equivalence classes. These equivalence classes are constructed so that elements and belong to the same equivalence class if, and only if, they are equivalent. Formally, given a set and an equivalence relation on the of an element in denoted by is the set of elements which are equivalent to It may be proven, from the defining properties of equivalence relations, that the equivalence classes form a partition of This partition—the set of equivalence classes—is sometimes called the quotient set or the quotient space of by and is denoted by .

Algebraic integer

In algebraic number theory, an algebraic integer is a complex number which is integral over the integers. That is, an algebraic integer is a complex root of some monic polynomial (a polynomial whose leading coefficient is 1) whose coefficients are integers. The set of all algebraic integers A is closed under addition, subtraction and multiplication and therefore is a commutative subring of the complex numbers. The ring of integers of a number field K, denoted by OK, is the intersection of K and A: it can also be characterised as the maximal order of the field K.

Quadratic integer

In number theory, quadratic integers are a generalization of the usual integers to quadratic fields. Quadratic integers are algebraic integers of degree two, that is, solutions of equations of the form x2 + bx + c = 0 with b and c (usual) integers. When algebraic integers are considered, the usual integers are often called rational integers. Common examples of quadratic integers are the square roots of rational integers, such as , and the complex number i = , which generates the Gaussian integers.

Gaussian integer

In number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as or Gaussian integers share many properties with integers: they form a Euclidean domain, and have thus a Euclidean division and a Euclidean algorithm; this implies unique factorization and many related properties. However, Gaussian integers do not have a total ordering that respects arithmetic.

Related lectures (43)

Rational Numbers: Construction and Properties

Covers the construction and properties of rational numbers, including fractions and equivalence, with proofs on commutativity.

Multiplication in Rational Numbers

Covers the extension of addition and multiplication operations in rational numbers.

Properties of Division

Covers the properties of division in integers and the relationship between divisibility and unique quotients.

Basic Properties MOOC: Analysis I (part 1): Prelude, basic concepts, real numbers MOOC: Analysis I

Covers basic properties of natural numbers, including order relations and inverses.

Linear Algebra: Reciprocity and Equivalence