divisor as factor of principal divisor
Let an integral domain have a divisor theory . The definition of divisor theory (http://planetmath.org/DivisorTheory) implies that for any divisor , there exists an element of such that divides the principal divisor , i.e. that with a divisor. The following theorem states that may always be chosen such that it is coprime with any beforehand given divisor.
Theorem. For any two divisors and , there is a principal divisor such that
Proof. Let all distinct prime divisors, which divide the product , and let the divisor be exactly divisible (http://planetmath.org/ExactlyDivides) by the powers (the cases are not excluded). For each , we choose a nonzero element of being exactly divisible by the power ; the choosing is possible, since any nonzero element of the ideal determined by the divisor , not belonging to the sub-ideal determined by the divisor , will do. According to the Chinese remainder theorem (http://planetmath.org/ChineseRemainderTheoremInTermsOfDivisorTheory), there exists a nonzero element of the ring such that
Because is divisible by , the element is divisible by , i.e. . If one of the divisors would divide , then would be divisible by and thus by (1), also were divisible by . Therefore, no one of the prime divisors divides . On the other hand, every prime divisor dividing the divisor divides and thus is one of . Accordingly, the divisors and have no common prime divisor, i.e. .
- 1 М. М. Постников: Введение в теорию алгебраических чисел. Издательство ‘‘Наука’’. Москва (1982).
|Title||divisor as factor of principal divisor|
|Date of creation||2013-03-22 18:02:09|
|Last modified on||2013-03-22 18:02:09|
|Last modified by||pahio (2872)|