# Number Theory Lecture 4

This lecture ( NumTh4 ) was given on 9/8/09

### Excerpts

To do: find a finite extension of number fields that isn’t Galois.

Continuing from last time with the frame work . Say a downstairs has primes upstairs, then if its a Galois extension, then

The action of the Galois group is transitive on the so the orbit has size m and basic group theory say . It then become relatively easy to prove that for , ; this is the inertia subgroup. Note is a finite extension of finite fields and is generated by the Frobenious morphisms , so it can be checked directly that its a Galois extension.

In the case then is an isomorphism and define . Now replacing with another prime lying over conjugates by an element of the Galois group: . So associate to the resulting conjugacy class

its called the Frobenius class at .

There was some stuff on the compositum of fields and as a result I have this question: If are all Galois, and if I take any , this clearly by restricting I get a map , why does this land in ?

Basically because everything is Galois: if then it satisfies some polynomial with coefficients in , so the action of permutes the roots of so sends to another element in ; both satisfy the same polynomial equation, so they are in the same extension field where .

Thm (Tchebotarev Density thm): Let be a Galois ext. of number fields. Let be a conjugacy class, then the set of primes such that has density .

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- Published:
- September 9, 2009 / 10:19 am

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- Number Theory (course)

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