# 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 $\frac{\alpha}{\beta}$ (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 $\frac{\alpha}{\beta}$ 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 $\frac{1}{\beta}$ 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 $\mathbb{A}$, 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 |