units of real cubic fields with exactly one real embedding
Let be a number field with such that has exactly one real embedding. Thus, and . Let denote the group of units of the ring of integers of . By Dirichlet’s unit theorem, since . The only roots of unity in are and because . Thus, . Therefore, there exists with , such that every element of is of the form for some .
Let and such that the conjugates of are and . Since is a unit, . Thus, . Since and , it must be the case that . Thus, . One can then deduce that . Since the maximum value of the polynomial is at most , one can deduce that . Define . Then . Thus, . From this, one can obtain that . (Note that a higher lower bound on is desirable, and the one stated here is much higher than that stated in Marcus.) Thus, . Therefore, if an element can be found such that , then .
Following are some applications:
The above can also be used for any number field with such that has exactly one real embedding. Let be the real embedding. Then the above produces the fundamental unit of . Thus, is a fundamental unit of .
- 1 Marcus, Daniel A. Number Fields. New York: Springer-Verlag, 1977.
|Title||units of real cubic fields with exactly one real embedding|
|Date of creation||2013-03-22 16:02:25|
|Last modified on||2013-03-22 16:02:25|
|Last modified by||Wkbj79 (1863)|