proof of norm and trace of algebraic number
Before proving this theorem, a lemma will be stated and proven.
Let be a number field with , such that , and and denote the absolute norm (http://planetmath.org/AbsoluteNorm) and absolute trace (http://planetmath.org/AbsoluteTrace) of , respectively. Then divides , and .
Note that divides because .
Note also that each of the embeddings of into extends to exactly embeddings of into . Thus,
Now, the above theorem will be proven.
Proof of theorem 1. Let be the minimal polynomial for over . Then , where is as in the previous lemma. Note that is equal to the absolute value of the constant term of and that is equal to the opposite of the coefficient of of . Thus, . Therefore, and . Moreover, if is an algebraic integer, then , , , and .
If , then , , and .
Finally, if , then
An algebraic integer is a unit if and only if its . Thus, in the minimal polynomial of an algebraic unit is always .
Let . Since is an algebraic integer, is finite. Let denote the ring of integers of .
If , then let be the minimal polynomial of over . Let such that . Then . Thus, . Since , it follows that is a unit in .
Conversely, let be a unit in . Let with . Since and , it follows that . ∎
- 1 Marcus, Daniel A. Number Fields. New York: Springer-Verlag, 1977.
|Title||proof of norm and trace of algebraic number|
|Date of creation||2013-03-22 15:58:53|
|Last modified on||2013-03-22 15:58:53|
|Last modified by||Wkbj79 (1863)|