Wholeness ring

In the mathematical branch of algebraic number theory , the whole ring of an algebraic number field is the analogue of the ring of whole numbers in the case of the field of rational numbers . The elements of a wholeness ring are called algebraic integers , the set of all algebraic integers is the wholeness ring in the field of all algebraic numbers .

definition

Let it be an algebraic number field , i.e. H. a finite extension of the field of rational numbers . Then the wholeness ring of is defined as the whole closure of in , i.e. H. the subset of those that have an equation of form ${\displaystyle K}$ ${\displaystyle {\mathcal {O}}_{K}}$ ${\displaystyle K}$ ${\displaystyle \mathbb {Z} }$ ${\displaystyle K}$ ${\displaystyle x\in K}$

${\displaystyle x^{n}+c_{n-1}x^{n-1}+\ldots +c_{1}x+c_{0}=0}$

meet with . Note that the coefficient of (the leading coefficient of the polynomial ) must be 1. Such polynomials are called normalized . Without these restrictions you would get the whole body . ${\displaystyle c_{i}\in \mathbb {Z} }$ ${\displaystyle x^{n}}$ ${\displaystyle x^{n}+c_{n-1}x^{n-1}+\ldots +c_{1}x+c_{0}}$ ${\displaystyle K}$

An equivalent definition reads: The wholeness ring of is the maximum order in the sense of inclusion , the main order on . ${\displaystyle K}$ ${\displaystyle K}$

properties

• ${\displaystyle {\mathcal {O}}_{K}}$is a finitely generated, free module of rank . ${\displaystyle \mathbb {Z} }$ ${\displaystyle [K\colon \mathbb {Q} ]}$
• ${\displaystyle {\mathcal {O}}_{K}}$is a dedekind ring .
• The unit group of is described by the Dirichlet theorem of units . ${\displaystyle {\mathcal {O}}_{K}}$

Examples

• Is , so is the ring of Eisenstein numbers ${\displaystyle K=\mathbb {Q} (\mathrm {i} {\sqrt {3}})}$ ${\displaystyle {\mathcal {O}}_{K}}$

${\displaystyle u+v\cdot {\frac {-1+\mathrm {i} {\sqrt {3}}}{2}}}$ With ${\displaystyle u,v\in \mathbb {Z} .}$

Such a number is the zero of the polynomial

${\displaystyle X^{2}-(2u-v)X+(u^{2}-uv+v^{2}).}$

Conversely, satisfies the polynomial equation ${\displaystyle x=a+b\mathrm {i} {\sqrt {3}}\in K}$

${\displaystyle x^{2}+px+q=0}$ With ${\displaystyle p,q\in \mathbb {Z} ,}$

so follows and . One can show that then and are integer, so is ${\displaystyle p=-2a}$ ${\displaystyle q=a^{2}+3b^{2}}$ ${\displaystyle a+b}$ ${\displaystyle 2b}$

${\displaystyle x=(a+b)+2b\cdot {\frac {-1+\mathrm {i} {\sqrt {3}}}{2}}}$

an Eisenstein number.

• Is , then is the ring of whole Gaussian numbers . ${\displaystyle K=\mathbb {Q} (\mathrm {i} )}$ ${\displaystyle {\mathcal {O}}_{K}}$ ${\displaystyle \mathbb {Z} [\mathrm {i} ]}$

• In general, a wholeness basis looks like this for the wholeness ring of (with whole and square-free): ${\displaystyle \mathbb {Q} ({\sqrt {d}})}$ ${\displaystyle d}$

${\displaystyle \{1,{\sqrt {d}}\}}$if congruent 2 or 3 mod 4 ${\displaystyle d}$

${\displaystyle \{1,{\frac {1+{\sqrt {d}}}{2}}\}}$if congruent 1 mod 4 ${\displaystyle d}$

• If a primitive -th root of unity denotes , the wholeness ring of the -th body of the circle is the same . ${\displaystyle \zeta }$ ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle \mathbb {Q} (\zeta )}$ ${\displaystyle \mathbb {Z} [\zeta ]}$

See also

• order