divisors in base field and finite extension field
Let be the quotient field of an integral domain which has the divisor theory . Let a finite extension, be the integral closure of in and the uniquely determined divisor theory of (see the parent entry (http://planetmath.org/DivisorTheoryInFiniteExtension)). We will study the of the divisor monoids and .
Any element of , which is a part of , determines a principal divisor and another . The (multiplicative) monoid is isomorphically embedded (via ) in the monoid . Because the units of the ring , which belong to , are all units of and because associates always determine the same principal divisor, the mentioned embedding defines an isomorphic mapping
from the monoid of the principal divisors of into the monoid of the principal divisors of . One has the
Theorem. There is one and only one isomorphism from the divisor monoid into the divisor monoid such that its restriction to the principal divisors of coincides with (1). Then there is the following commutative diagram:
The isomorphism is determined as follows. Let be an arbitrary prime divisor in and the corresponding exponent valuation of the field . Let be the continuations of the exponent to , which correspond to the prime divisors in . If are the ramification indices of the exponents with respect to , then we have
Thus apparently, the factor of the principal divisor , which corresponds to the factor of the principal divisor , is . Then is settled by
When one identifies with its isomorphic image , we can write
i.e. the prime divisors in don’t in general remain as prime divisors in . On grounds of the identification one may speak of the divisibility of the divisors of by the divisors of . The coprime divisors of are coprime also as divisors of .
- 1 S. Borewicz & I. Safarevic: Zahlentheorie. Birkhäuser Verlag. Basel und Stuttgart (1966).
|Title||divisors in base field and finite extension field|
|Date of creation||2013-03-22 18:01:35|
|Last modified on||2013-03-22 18:01:35|
|Last modified by||pahio (2872)|