Let be a ring homomorphism. Let be a prime ideal, with . The algebra map induces an module structure on the ring . If the dimension of as an module exists, then it is called the inertial degree of over .
A particular case of special importance in number theory is when is a field extension and is the inclusion map of the ring of integers. In this case, the domain is a field, so is guaranteed to exist, and the inertial degree of over is denoted . We have the formula
where is the ramification index of over and the sum is taken over all prime ideals of dividing . The prime (and also the prime ) is said to be inert if .
Let be the inclusion of the integers into the Gaussian integers. A prime in may or may not factor in ; if it does factor, then it must factor as for some integers . Thus a prime factors into two primes if it equals , and remains prime in otherwise. There are then three categories of primes in :
For primes , the prime factors into the product of the two primes , with ramification index and inertial degree one.
For primes , the prime remains prime in and is a two dimensional field extension of , so the inertial degree is two and the ramification index is one.
1 Local interpretations & generalizations
For any extension of Dedekind domains, the inertial degree of the prime over the prime is equal to the inertial degree of over in the localizations at and . Moreover, the same is true even if we pass to completions of the local rings and at and . The preservation of inertial degree and ramification indices with respect to localization is one of the reasons why the technique of localization is a useful tool in the study of such domains.
As in the case of ramification indices, it is possible to define the notion of inertial degree in the more general setting of locally ringed spaces. However, the generalizations of inertial degree are not as widely used because in algebraic geometry one usually works with a fixed base field, which makes all the residue fields at the points equal to the same field.
|Date of creation||2013-03-22 12:38:17|
|Last modified on||2013-03-22 12:38:17|
|Last modified by||djao (24)|