splitting and ramification in number fields and Galois extensions
Let be an extension of number fields and let and be their respective rings of integers. The ring of integers of a number field is a Dedekind domain, and these enjoy the property that every ideal factors uniquely as a finite product of prime ideals (see the entry fractional ideal (http://planetmath.org/FractionalIdeal)). Let be a prime ideal of . Then is an ideal of . Let us assume that the prime ideal factorization of into primes of is as follows:
We say that the primes lie above and (divides). The exponent (commonly denoted as ) is the ramification index of over . Notice that for each prime ideal , the quotient ring is a finite field extension of the finite field (also called the residue field). The degree of this extension is called the inertial degree of over and it is usually denoted by:
Notice that as it is pointed out in the entry “inertial degree (http://planetmath.org/InertialDegree)”, the ramification index and the inertial degree are related by the formula:
Let and be as above.
If for some , then we say that is ramified over and ramifies in . If for all then we say that is unramified in .
On the other hand, if for all , we say that is totally split (or splits completely) in . Notice that there are exactly prime ideals of lying above .
Let be the characteristic of the residue field . If and and are relatively prime, then we say that is tamely ramified. If then we say that is strongly ramified (or wildly ramified).
Assume that is a Galois extension of number fields. Then all the ramification indices are equal to the same number , all the inertial degrees are equal to the same number and the ideal factors as:
|Title||splitting and ramification in number fields and Galois extensions|
|Date of creation||2013-03-22 15:05:29|
|Last modified on||2013-03-22 15:05:29|
|Last modified by||alozano (2414)|