Dirichletscher unit set

The Dirichlet set of units named after Peter Gustav Lejeune Dirichlet is one of the first results of algebraic number theory . The sentence describes the structure of the unit group of the whole ring of an algebraic number field .

formulation

Let it be an algebraic number field and its wholeness ring . Then the unit group is finitely generated and the rank of its free part is equal ${\displaystyle K}$ ${\displaystyle {\mathcal {O}}_{K}}$ ${\displaystyle {\mathcal {O}}_{K}^{\times }}$

${\displaystyle r+s-1.}$

Here is the number of embeddings and the number of pairs of complex-conjugated embeddings (which are not real embeddings). So it applies . If the extension is Galois , then is or equal . ${\displaystyle r}$ ${\displaystyle K\to \mathbb {R} }$ ${\displaystyle s}$ ${\displaystyle K\to \mathbb {C} }$ ${\displaystyle [K:\mathbb {Q} ]=r+2s}$ ${\displaystyle K/\mathbb {Q} }$ ${\displaystyle r}$ ${\displaystyle s}$ ${\displaystyle 0}$

The torsion component of the unit group is the group of unit roots in . ${\displaystyle K}$

Evidence sketch in a special case

Let it be (so we already choose a real embedding). Then is , and the unit group ${\displaystyle K=\mathbb {Q} ({\sqrt {2}})\subset \mathbb {R} }$ ${\displaystyle {\mathcal {O}}_{K}=\mathbb {Z} [{\sqrt {2}}]}$

${\displaystyle {\mathcal {O}}_{K}^{\times }=\{x+y{\sqrt {2}}\mid x,y\in \mathbb {Z} ,\quad x^{2}-2y^{2}=\pm 1\}.}$

(The equation is called " Pell's equation ".) ${\displaystyle x^{2}-dy^{2}=\pm 1}$

In this case is and . The Dirichlet theorem of units therefore predicts that the rank is equal to 1. ${\displaystyle r=2}$ ${\displaystyle s=0}$ ${\displaystyle {\mathcal {O}}_{K}^{\times }}$

For example, since there is a unit that is not a unit root, the rank must be at least 1. If the rank were greater, then there could be no discrete subgroup of , and one knows that a subgroup of is either discrete or dense . So there would be a unit that is "approximately" 1. But now and are two numbers, the product of which is, so if one of them is approximately 1, then the other is approximately . On the other hand, they differ by the number that is “significantly” larger than the distance between and , if is. But if it is , then it is obvious that we only get the roots of unity . ${\displaystyle 3+2{\sqrt {2}}}$ ${\displaystyle {\mathcal {O}}_{K}^{\times }\cap \mathbb {R} _{>0}^{\times }}$ ${\displaystyle \mathbb {R} _{>0}^{\times }}$ ${\displaystyle \mathbb {R} _{>0}^{\times }}$ ${\displaystyle x+y{\sqrt {2}}}$ ${\displaystyle x-y{\sqrt {2}}}$ ${\displaystyle \pm 1}$ ${\displaystyle \pm 1}$ ${\displaystyle 2y{\sqrt {2}}}$ ${\displaystyle 1}$ ${\displaystyle \pm 1}$ ${\displaystyle y\neq 0}$ ${\displaystyle y=0}$ ${\displaystyle x=\pm 1}$ ${\displaystyle \pm 1\in {\mathcal {O}}_{K}^{\times }}$