divisor theory and exponent valuations
A divisor theory of an integral domain determines via its prime divisors a certain set of exponent valuations on the quotient field of . Assume to be known this set of exponents (http://planetmath.org/ExponentValuation2) corresponding the prime divisors . There is a bijective correspondence between the elements of and of the set of all prime divisors. The set of the prime divisors determines completely the of the free monoid of all divisors in question. The homomorphism is then defined by the condition
since for any element of there exists only a finite number of exponents which do not vanish on (corresponding the different prime divisor factors (http://planetmath.org/DivisibilityInRings) of the principal divisor ).
Theorem. Let be an integral domain with quotient field and a given set of exponents (http://planetmath.org/ExponentValuation2) of . The exponents in determine, as in (1), a divisor theory of iff the following three conditions are in :
For every there is at most a finite number of exponents such that .
An element belongs to if and only if for each .
For the proof of the theorem, we mention only how to construct the divisors when we have the exponent set fulfilling the three conditions of the theorem. We choose a commutative monoid that allows unique prime factorisation and that may be mapped bijectively onto . The exponent in which corresponds to arbitrary prime element is denoted by . Then we obtain the homomorphism
which can be seen to satisfy all required properties for a divisor theory .
- 1 S. Borewicz & I. Safarevic: Zahlentheorie. Birkhäuser Verlag. Basel und Stuttgart (1966).
|Title||divisor theory and exponent valuations|
|Date of creation||2013-03-22 17:59:34|
|Last modified on||2013-03-22 17:59:34|
|Last modified by||pahio (2872)|
|Synonym||divisors and exponents|