units of quadratic fields
Dirichlet’s unit theorem gives all units of an algebraic number field in the unique form
where is a primitive root of unity in , the ’s are the fundamental units of , , , .
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The case of a real quadratic field , the square-free : , , . So we obtain
because is the only real primitive root of unity (). Thus, every real quadratic field has infinitely many units and a unique fundamental unit .
Examples: If , then ; if , then .
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The case of any imaginary quadratic field ; here , the square-free : The conjugates of are the pure imaginary numbers , hence , , . Thus we see that all units are
1) . The field contains the primitive fourth root of unity, e.g. , and therefore all units in the field are , where .
2) . 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
hence all units are , where , or, equivalently, , where .
3) , . The only roots of unity in the field are ; hence , , and the units of the field are simply , where .
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 |