# 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  $\beta\neq 0$) 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 $x^{m}+b_{1}x^{m-1}+\ldots+b_{m-1}x+b_{m}$ is the minimal polynomial of $\beta$, the equation  $\beta^{m}+b_{1}\beta^{m-1}+\ldots+b_{m-1}\beta+b_{m}=0$  implies

 $\left(\frac{1}{\beta}\right)^{m}\!+\frac{b_{m-1}}{b_{m}}\!\left(\frac{1}{\beta% }\right)^{m-1}\!+\ldots+\frac{b_{1}}{b_{m}}\!\cdot\!\frac{1}{\beta}+\frac{1}{b% _{m}}\;=\;0.$

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

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 $\mathbb{A}$, 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 $\mathbb{A}$).

 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