field of algebraic numbers

As special cases of the theorem of the parent “polynomial equation with algebraic coefficients (” of this entry, one obtains the

Corollary.  If α and β are algebraic numbersMathworldPlanetmath, then also α+β, α-β, αβ and αβ (provided  β0) are algebraic numbers.  If α and β are algebraic integersMathworldPlanetmath, then also α+β, α-β and αβ are algebraic integers.

The case of αβ needs an additional consideration:  If xm+b1xm-1++bm-1x+bm is the minimal polynomial of β, the equation  βm+b1βm-1++bm-1β+bm=0  implies

(1β)m+bm-1bm(1β)m-1++b1bm1β+1bm= 0.

Hence 1β is an algebraic number, and therefore also α1β.

It follows from the corollary that the set of all algebraic numbers is a field and the set of all algebraic integers is a ring (an integral domain, too).  Moreover, the mentioned theorem implies that the field of algebraic numbers is algebraically closedMathworldPlanetmath and the ring of algebraic integers integrally closed.  The field of algebraic numbers, which is sometimes denoted by 𝔸, contains for example the complex numbersMathworldPlanetmathPlanetmath obtained from rational numbersPlanetmathPlanetmathPlanetmath by using arithmetic operations and taking (these numbers form a subfieldMathworldPlanetmath of 𝔸).

Title field of algebraic numbers
Canonical name FieldOfAlgebraicNumbers
Date of creation 2015-11-18 14:30:41
Last modified on 2015-11-18 14:30:41
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 14
Author pahio (2872)
Entry type Definition
Classification msc 11R04
Related topic AlgebraicSumAndProduct
Related topic SubfieldCriterion
Related topic AlgebraicNumbersAreCountable
Related topic RingWithoutIrreducibles
Related topic AllAlgebraicNumbersInASequence
Defines ring of algebraic integers