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[parent] fundamental units (Definition)

The ring $ R$ of algebraic integers of any algebraic number field contains a finite set $ H = \{\eta_1,\, \eta_2,\, \ldots,\, \eta_t\}$ of so-called fundamental units such that every unit $ \varepsilon$ of $ R$ is a power product of these, multiplied by a root of unity:

$\displaystyle \varepsilon = \zeta\!\cdot\!\eta_1^{k_1}\eta_2^{k_2}\ldots\eta_t^{k_t}$
Conversely, every such element $ \varepsilon$ of the field is a unit of $ R$.

Examples: units of quadratic fields, units of certain cubic fields

For some algebraic number fields, such as all imaginary quadratic fields, the set $ H$ may be empty ($ t = 0$). In the case of a single fundamental unit ($ t = 1$), which occurs e.g. in all real quadratic fields, there are two alternative units $ \eta$ and its conjugate $ \overline{\eta}$ which one can use as fundamental unit; then we can speak of the uniquely determined fundamental unit $ \eta_1$ which is greater than 1.



"fundamental units" is owned by pahio.
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See Also: number field, algebraic integer

Keywords:  Dirichlet's unit theorem

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table of some fundamental units (Result) by pahio
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Cross-references: conjugate, real quadratic fields, imaginary, units of quadratic fields, field, root of unity, product, unit, finite set, contains, algebraic number field, algebraic integers, ring
There are 5 references to this entry.

This is version 18 of fundamental units, born on 2004-08-06, modified 2006-10-16.
Object id is 6080, canonical name is FundamentalUnits.
Accessed 2387 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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