units of quadratic fields

Dirichlet’s unit theorem gives all units of an algebraic number field $\mathbb{Q}(\vartheta )$ in the unique form

$$\epsilon ={\zeta }^n{\eta }_1^{k_1}{\eta }_2^{k_2}\mathrm{\dots }{\eta }_t^{k_t},$$

where $\zeta$ is a primitive $w^{\mathrm{th}}$ root of unity in $\mathbb{Q}(\vartheta )$, the ${\eta }_j$’s are the fundamental units of $\mathbb{Q}(\vartheta )$,  $0\leqq n\leqq w-1$,  $k_j\in \mathbb{Z}$  $\forall j$,  $t=r+s-1$.

• The case of a real quadratic field $\mathbb{Q}(\sqrt{m})$, the square-free  $m>1$:  $r=2$,  $s=0$,  $t=r+s-1=1$.  So we obtain

  $$\epsilon ={\zeta }^n{\eta }^k=\pm {\eta }^k,$$

  because  $\zeta =-1$  is the only real primitive root of unity ($w=2$).  Thus, every real quadratic field has infinitely many units and a unique fundamental unit $\eta$.

  Examples:  If  $m=3$,  then  $\eta =2+\sqrt{3}$;  if  $m=421$,  then  $\eta =\frac{444939+21685\sqrt{421}}{2}$.

• The case of any imaginary quadratic field $\mathbb{Q}(\vartheta )$; here  $\vartheta =\sqrt{m}$,  the square-free  :  The conjugates of $\vartheta$ are the pure imaginary numbers $\pm \sqrt{m}$, hence  $r=0$,  $2s=2$,  $t=r+s-1=0$.  Thus we see that all units are

  $$\epsilon ={\zeta }^n.$$

  1) $m=-1$.  The field contains the primitive fourth root of unity, e.g. $i$, and therefore all units in the field $\mathbb{Q}(i)$ are $i^n$, where  $n=0,\mathrm{\hspace{0.17em}1},\mathrm{\hspace{0.17em}2},\mathrm{\hspace{0.17em}3}$.

  2) $m=-3$.  The field in question is a cyclotomic field (http://planetmath.org/CyclotomicExtension) containing the primitive third root of unity and also the primitive sixth root of unity, namely

  $$\zeta =\cos \frac{2\pi }{6}+i\sin \frac{2\pi }{6};$$

  hence all units are  $\epsilon ={(\frac{1+\sqrt{-3}}{2})}^n$,  where  $n=0,\mathrm{\hspace{0.17em}1},\mathrm{\dots },\mathrm{\hspace{0.17em}5}$, or, equivalently,  $\epsilon =\pm {(\frac{-1+\sqrt{-3}}{2})}^n$,  where $n=0,\mathrm{\hspace{0.17em}1},\mathrm{\hspace{0.17em}2}$.

  3) $m=-2$,  .   The only roots of unity in the field are $\pm 1$; hence  $\zeta =-1$,  $w=2$,  and the units of the field are simply ${(-1)}^n$, where  $n=0,\mathrm{\hspace{0.17em}1}$.

Title	units of quadratic fields
Canonical name	UnitsOfQuadraticFields
Date of creation	2013-03-22 14:16:30
Last modified on	2013-03-22 14:16:30
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	39
Author	pahio (2872)
Entry type	Application
Classification	msc 11R04
Classification	msc 11R27
Synonym	quadratic unit
Related topic	Unit
Related topic	NumberField
Related topic	ImaginaryQuadraticField
Related topic	SomethingRelatedToFundamentalUnits