**Are you an EPFL student looking for a semester project?**

Work with us on data science and visualisation projects, and deploy your project as an app on top of GraphSearch.

Lecture# Field Properties: Irreducibility and Units

Description

This lecture discusses the properties of fields, focusing on irreducibility and units. It covers the definition of a field, prime elements, Euclidean domains, irreducible elements, maximal ideals, and the relationship between fields and maximal ideals. The lecture also explores the concept of irreducibility in polynomials and the conditions for a polynomial to be considered irreducible.

Official source

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

In course

Instructor

Related concepts (53)

MATH-310: Algebra

This is an introduction to modern algebra: groups, rings and fields.

Prime ideal

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 integers are the sets that contain all the multiples of a given prime number, together with the zero ideal. Primitive ideals are prime, and prime ideals are both primary and semiprime. An ideal P of a commutative ring R is prime if it has the following two properties: If a and b are two elements of R such that their product ab is an element of P, then a is in P or b is in P, P is not the whole ring R.

Maximal ideal

In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all proper ideals. In other words, I is a maximal ideal of a ring R if there are no other ideals contained between I and R. Maximal ideals are important because the quotients of rings by maximal ideals are simple rings, and in the special case of unital commutative rings they are also fields.

Chebyshev polynomials

The Chebyshev polynomials are two sequences of polynomials related to the cosine and sine functions, notated as and . They can be defined in several equivalent ways, one of which starts with trigonometric functions: The Chebyshev polynomials of the first kind are defined by Similarly, the Chebyshev polynomials of the second kind are defined by That these expressions define polynomials in may not be obvious at first sight, but follows by rewriting and using de Moivre's formula or by using the angle sum formulas for and repeatedly.

Euclidean domain

In mathematics, more specifically in ring theory, a Euclidean domain (also called a Euclidean ring) is an integral domain that can be endowed with a Euclidean function which allows a suitable generalization of the Euclidean division of integers. This generalized Euclidean algorithm can be put to many of the same uses as Euclid's original algorithm in the ring of integers: in any Euclidean domain, one can apply the Euclidean algorithm to compute the greatest common divisor of any two elements.

Laguerre polynomials

In mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834–1886), are solutions of Laguerre's differential equation: which is a second-order linear differential equation. This equation has nonsingular solutions only if n is a non-negative integer. Sometimes the name Laguerre polynomials is used for solutions of where n is still a non-negative integer. Then they are also named generalized Laguerre polynomials, as will be done here (alternatively associated Laguerre polynomials or, rarely, Sonine polynomials, after their inventor Nikolay Yakovlevich Sonin).

Related lectures (196)

Finite Fields: Construction and PropertiesMATH-310: Algebra

Explores the construction and properties of finite fields, including irreducible polynomials and the Chinese Remainder Theorem.

Properties of Euclidean DomainsMATH-310: Algebra

Covers the properties of Euclidean domains and irreducible elements in polynomial rings.

Rings and Fields: Principal Ideals and Ring HomomorphismsMATH-310: Algebra

Covers principal ideals, ring homomorphisms, and more in commutative rings and fields.

Polynomials, Division, and Ideals

Explores polynomials, their operations, and the concept of ideals in polynomial rings.

Chinese Remainder Theorem: Rings and FieldsMATH-310: Algebra

Covers the Chinese remainder theorem for commutative rings and integers, polynomial rings, and Euclidean domains.