field of algebraic numbers

As special cases of the theorem of the parent “polynomial equation with algebraic coefficients (http://planetmath.org/polynomialequationwithalgebraiccoefficients)” of this entry, one obtains the

Corollary.  If $\alpha$ and $\beta$ are algebraic numbers, then also $\alpha +\beta$, $\alpha -\beta$, $\alpha \beta$ and ${\displaystyle \frac{\alpha }{\beta }}$ (provided  $\beta \ne 0$) are algebraic numbers.  If $\alpha$ and $\beta$ are algebraic integers, then also $\alpha +\beta$, $\alpha -\beta$ and $\alpha \beta$ are algebraic integers.

The case of ${\displaystyle \frac{\alpha }{\beta }}$ needs an additional consideration:  If $x^m+b_1{x}^{m-1}+\mathrm{\dots }+b_{m-1}x+b_m$ is the minimal polynomial of $\beta$, the equation  ${\beta }^m+b_1{\beta }^{m-1}+\mathrm{\dots }+b_{m-1}\beta +b_m=0$  implies

$${\left(\frac{1}{\beta }\right)}^m+\frac{b_{m-1}}{b_m}{\left(\frac{1}{\beta }\right)}^{m-1}+\mathrm{\dots }+\frac{b_1}{b_m}\cdot \frac{1}{\beta }+\frac{1}{b_m}=\mathrm{\hspace{0.33em}0}.$$

Hence ${\displaystyle \frac{1}{\beta }}$ is an algebraic number, and therefore also $\alpha \cdot {\displaystyle \frac{1}{\beta }}$.

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 closed and the ring of algebraic integers integrally closed.  The field of algebraic numbers, which is sometimes denoted by $𝔸$, contains for example the complex numbers obtained from rational numbers by using arithmetic operations and taking http://planetmath.org/node/5667roots (these numbers form a subfield 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