Definitions
Let L / K be a finite extension of number fields, and let B and A be the corresponding ring of integers of L and K, respectively, which are defined to be the integral closure of the integers Z in the field in question.
Finally, let p be a non-zero prime ideal in A, or equivalently, a maximal ideal, so that the residue A/p is a field.
From the basic theory of one-dimensional rings follows the existence of a unique decomposition
of the ideal pB generated in B by p into a product of distinct maximal ideals Pj, with multiplicities e(j).
The multiplicity e(j) are called ramification indices of the extension at p. If they are all equal to 1 and if in addition the field extensions B/Pj over A/p is separable, the field extension L/K is called unramified at p.
If this is the case, by the Chinese remainder theorem, the quotient
-
- B/pB
- is a product of fields
- Fj = B/Pj.
Read more about this topic: Splitting Of Prime Ideals In Galois Extensions
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