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[parent] units of quadratic fields (Application)

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

$\displaystyle \varepsilon = \zeta^{n}\eta_1^{k_1}\eta_2^{k_2}...\eta_t^{k_t},$
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
    $\displaystyle \varepsilon = \zeta^{n}\eta^{k} = \pm\eta^{k},$
    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 $ m < 0$: 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
    $\displaystyle \varepsilon = \zeta^{n}.$

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

    2) $ m = -3$. The field in question is a cyclotomic field containing the primitive third root of unity and also the primitive sixth root of unity, namely

    $\displaystyle \zeta = \cos{\frac{2\pi}{6}}+i\sin{\frac{2\pi}{6}};$
    hence all units are $ \varepsilon = (\frac{1+\sqrt{-3}}{2})^{n}$, where $ n = 0,\,1,\,\ldots,\,5$, or, equivalently, $ \varepsilon = \pm(\frac{-1+\sqrt{-3}}{2})^{n}$, where $ n = 0,\,1,\,2$.

    3) $ m = -2$, $ m <-3$. 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,\,1$.



"units of quadratic fields" is owned by pahio.
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See Also: unit, number field, imaginary quadratic field, table of some fundamental units

Other names:  quadratic unit

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Cross-references: contains, field, pure imaginary numbers, conjugates, imaginary quadratic field, primitive root of unity, real, square-free, real quadratic field, fundamental units, root of unity, algebraic number field, units, Dirichlet's unit theorem
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This is version 36 of units of quadratic fields, born on 2004-03-21, modified 2008-04-09.
Object id is 5726, canonical name is UnitsOfQuadraticFields.
Accessed 3707 times total.

Classification:
AMS MSC: 11R04 (Number theory :: Algebraic number theory: global fields :: Algebraic numbers; rings of algebraic integers)
 11R27 (Number theory :: Algebraic number theory: global fields :: Units and factorization)
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