# Field (mathematics)

In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do. A field is thus a fundamental algebraic structure which is widely used in algebra, number theory, and many other areas of mathematics. The best known fields are the field of rational numbers, the field of real numbers and the field of complex numbers. Many other fields, such as fields of rational functions, algebraic function fields, algebraic number fields, and p-adic fields are commonly used and studied in mathematics, particularly in number theory and algebraic geometry. Most cryptographic protocols rely on finite fields, i.e., fields with finitely many elements. The relation of two fields is expressed by the notion of a field extension. Galois theory, initiated by Évariste Galois in the 1830s, is devoted to understanding the symmetries of field extensions. Among other results, this theory shows that angle trisection and squaring the circle cannot be done with a compass and straightedge. Moreover, it shows that quintic equations are, in general, algebraically unsolvable. Fields serve as foundational notions in several mathematical domains. This includes different branches of mathematical analysis, which are based on fields with additional structure. Basic theorems in analysis hinge on the structural properties of the field of real numbers. Most importantly for algebraic purposes, any field may be used as the scalars for a vector space, which is the standard general context for linear algebra. Number fields, the siblings of the field of rational numbers, are studied in depth in number theory. Function fields can help describe properties of geometric objects. (Wikipedia).

Field Definition (expanded) - Abstract Algebra

The field is one of the key objects you will learn about in abstract algebra. Fields generalize the real numbers and complex numbers. They are sets with two operations that come with all the features you could wish for: commutativity, inverses, identities, associativity, and more. They

From playlist Abstract Algebra

Field Theory: Definition/ Axioms

This video is about the basics axioms of fields.

From playlist Basics: Field Theory

What is a field ?

Definition of a Field In this video, I define the concept of a field, which is basically any set where you can add, subtract, add, and divide things. Then I show some neat properties that have to be true in fields. Enjoy! What is an Ordered Field: https://youtu.be/6mc5E6x7FMQ Check out

From playlist Real Numbers

Abstract Algebra: The definition of a Field

Learn the definition of a Field, one of the central objects in abstract algebra. We give several familiar examples and a more unusual example. ♦♦♦♦♦♦♦♦♦♦ Ways to support our channel: ► Join our Patreon : https://www.patreon.com/socratica ► Make a one-time PayPal donation: https://www

From playlist Abstract Algebra

The Structure of Fields: What is a field and a connection between groups and fields

This video is primarily meant to help develop some ideas around the structure of fields and a connection between groups and fields (which will allow me to create more abstract algebra videos in the future! 😀😅🤓) 00:00 Intro 01:04 What is a Field? Here we give the definition of a field in

From playlist The New CHALKboard

Careers in STEM : Why They’re Important

Hey students! A career in applied mathematics isn't just about crunching numbers. It's a path that uses mathematics to solve real-world problems in important, ever-changing areas. But the landscape for this field is evolving and good grades aren't enough to get you where you want to be. Wh

From playlist What is math used for?

Field Theory -- Qbar, the field of algebraic numbers -- Lecture 8

In this video we show that QQbar, the algebraic closure of the rational numbers is countable.

From playlist Field Theory

Field Theory: Polynomials

This video is about polynomials with coefficients in a field.

From playlist Basics: Field Theory

Fields - Field Theory - Lecture 00

This is the first in a series of videos for my abstract algebra class during the 2020 shutdown. This lecture is intended to rapidly catch students up who are going to follow online and aren't from UVM. We are using Dummit and Foote.

From playlist Field Theory

PUBLIC OPENING featuring Cédric Villani: The Many Facets of Entropy [2014]

Video taken from: http://www.fields.utoronto.ca/programs/scientific/fieldsmedalsym/14-15/

From playlist Mathematics

We DON'T Understand Magnetism (According to Quantum Mechanics) - Aharonov-Bohm Effect by Parth G

The first 1000 people to use the link will get a free trial of Skillshare Premium Membership: https://skl.sh/parthg06211 Scientists have often thought that magnetic (and electric) fields are fundamental quantities that relate to real, physical, observable things in the universe. And they

From playlist Quantum Physics by Parth G

Quantum Field Theory as the Language of Physics by Nathan Seiberg

ICTS at Ten ORGANIZERS: Rajesh Gopakumar and Spenta R. Wadia DATE: 04 January 2018 to 06 January 2018 VENUE: International Centre for Theoretical Sciences, Bengaluru This is the tenth year of ICTS-TIFR since it came into existence on 2nd August 2007. ICTS has now grown to have more tha

From playlist ICTS at Ten

The Potential to Make Electric Fields Easier to Deal With | Electromagnetism by Parth G

Some mathematical identities combined with Maxwell's equations allow us to define electric and magnetic potentials... but why are they useful? Hi everyone! In a recent video, I talked about how the magnetic vector potential was a different way to view magnetic fields, and why Quantum Mech

From playlist Classical Physics by Parth G

Abel Prize award ceremony 2021

The ceremony honours both the 2020-winners, Hillel Furstenberg and Gregory Margulis, and the 2021-winners, Avi Wigderson and László́ Lovász. 0:30 Haddy N'jie sings Feeling Good 3:18 Welcome by Master of ceremonies, Haddy N'jie 4:46 On the nomination process and the work of the Abel Prize

From playlist Gregory Margulis

Quantum Field Theory and Quantum Topology — Jørgen Andersen

Serious Science - http://serious-science.org Mathematician Jørgen Andersen on topological quantum field theory, path integrals, and quantization of moduli spaces http://serious-science.org/quantum-field-theory-and-quantum-topology%E2%80%A8-6301

From playlist Quantum Field Theory

Vincent Lafforgue - 2019 Breakthrough Prize in Mathematics Laureate

Vincent Lafforgue CNRS and Institut Fourier, Université Grenoble Alpes 2019 Breakthrough Prize in Mathematics For ground breaking contributions to several areas of mathematics, in particular to the Langlands program in the function field case. https://breakthroughprize.org/Laureates/3/L3

From playlist Mathematics

The Genesis and Transformations of General Relativity - J. Renn - 3/10/2016

Bacon Award Public Lecture: - “The Genesis and Transformations of General Relativity” by Jürgen Renn, Director, Max Planck Institute for the History of Science - Introduction by Jed Z. Buchwald, Doris and Henry Dreyfuss Professor of History, Caltech Learn more about General Relativity at

From playlist Research & Science

History of Science and Technology Q&A (December 29, 2021)

Stephen Wolfram hosts a live and unscripted Ask Me Anything about the history of science and technology for all ages. Find the playlist of Q&A's here: https://wolfr.am/youtube-sw-qa Originally livestreamed at: https://twitch.tv/stephen_wolfram 0:00 Start stream 1:00 SW begins talking

Analyzing Sets of Data: Range, Mean, Median, and Mode

What does the word "average" mean to you? There are a lot ways we use that word, and even in math, it can imply a few different things. If we want to summarize a set of data in a meaningful way, we can talk about the mean, the median, or the mode, and one might be more useful than another

From playlist Mathematics (All Of It)

John Tate - The Abel Prize interview 2010

0:00 Glimpses of the Abel Prize ceremony [In Norwegian] 0:23 Speech by Nils Christian Stenseth, President of the Norwegian Academy of Science and Letters [In Norwegian] 1:15 Tate Receives the Abel Prize from His Majesty King Harald V of Norway 1:41 Interview start [English]. Your father wa

From playlist John T. Tate