**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.

Concept# Cubic field

Summary

In mathematics, specifically the area of algebraic number theory, a cubic field is an algebraic number field of degree three.
If K is a field extension of the rational numbers Q of degree [K:Q] = 3, then K is called a cubic field. Any such field is isomorphic to a field of the form
where f is an irreducible cubic polynomial with coefficients in Q. If f has three real roots, then K is called a totally real cubic field and it is an example of a totally real field. If, on the other hand, f has a non-real root, then K is called a complex cubic field.
A cubic field K is called a cyclic cubic field if it contains all three roots of its generating polynomial f. Equivalently, K is a cyclic cubic field if it is a Galois extension of Q, in which case its Galois group over Q is cyclic of order three. This can only happen if K is totally real. It is a rare occurrence in the sense that if the set of cubic fields is ordered by discriminant, then the proportion of cubic fields which are cyclic approaches zero as the bound on the discriminant approaches infinity.
A cubic field is called a pure cubic field if it can be obtained by adjoining the real cube root of a cube-free positive integer n to the rational number field Q. Such fields are always complex cubic fields since each positive number has two complex non-real cube roots.
Adjoining the real cube root of 2 to the rational numbers gives the cubic field . This is an example of a pure cubic field, and hence of a complex cubic field. In fact, of all pure cubic fields, it has the smallest discriminant (in absolute value), namely −108.
The complex cubic field obtained by adjoining to Q a root of x3 + x2 − 1 is not pure. It has the smallest discriminant (in absolute value) of all cubic fields, namely −23.
Adjoining a root of x3 + x2 − 2x − 1 to Q yields a cyclic cubic field, and hence a totally real cubic field. It has the smallest discriminant of all totally real cubic fields, namely 49.
The field obtained by adjoining to Q a root of x3 + x2 − 3x − 1 is an example of a totally real cubic field that is not cyclic.

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.

Related publications (36)

Related people (11)

Related concepts (1)

Related courses (2)

Related lectures (20)

Algebraic number field

In mathematics, an algebraic number field (or simply number field) is an extension field of the field of rational numbers such that the field extension has finite degree (and hence is an algebraic field extension). Thus is a field that contains and has finite dimension when considered as a vector space over . The study of algebraic number fields, and, more generally, of algebraic extensions of the field of rational numbers, is the central topic of algebraic number theory.

MATH-317: Algebra V - Galois theory

Galois theory aims at describing the algebraic symmetries of fields. After reviewing the basic material (from the 2nd year course "Ring and Fields") and in particular the Galois correspondence, we wi

EE-201: Electromagnetics II : field computation

Ce cours traite de l'électromagnétisme dans le vide et dans les milieux continus. A partir des principes fondamentaux de l'électromagnétisme, on établit les méthodes de résolution des équations de Max

Equidistribution of CM Points

Explores the joint equidistribution of CM points for cubic fields, emphasizing periodic orbits and quadratic algebraic forms.

Decomposition & Inertia: Group Actions and Galois Theory

Explores decomposition groups, inertia subgroups, Galois theory, unramified primes, and cyclotomic fields in group actions and field extensions.

Linear Algebra in Dirac Notation

Covers linear algebra in Dirac notation, focusing on vector spaces and quantum bits.

In this text, we will show the existence of lattice packings in a family of dimensions by employing division algebras. This construction is a generalization of Venkatesh's lattice packing result Venkatesh (Int Math Res Notices 2013(7): 1628-1642, 2013). In ...

,

Let k be a field, and let L be an etale k-algebra of finite rank. If a is an element of k(x), let X-a be the affine variety defined by N-L/k(x) = a. Assuming that L has at least one factor that is a cyclic field extension of k, we give a combinatorial desc ...

Let K be a totally real number field of degree n >= 2. The inverse different of K gives rise to a lattice in Rn. We prove that the space of Schwartz Fourier eigenfunctions on R-n which vanish on the "component-wise square root" of this lattice, is infinite ...