First we define the different divisor of an extension of function fields. Let be a function field over a field and let be a finite separable extension of . Let be a prime of , i.e. a discrete valuation ring with , maximal ideal and quotient field equal to . Let be the integral closure of in . Notice that if is a prime ideal of , then the localization is a prime of (which is said to be lying over ). The maximal ideal of is .
Let be any prime of , then it lays over some prime ideal of and in fact, if then . Let be the exact power of dividing the different of over (the different of an extension of Dedekind domains is a fractional ideal). We define the different divisor of as follows:
as an element of the free abelian group generated by the prime ideals of .
|Date of creation||2013-03-22 15:34:40|
|Last modified on||2013-03-22 15:34:40|
|Last modified by||alozano (2414)|
|Defines||different divisor of an extension of function fields|