# Number Theory lecture 3

This lecture (lecture3) was given on September 3rd 2009.

### Excerpts:

The lecture explain why the following story is true. Consider an extension of number fields . There is also a map on the ring of integers . The convention is is prime upstairs and . Its a standard result that is prime. Let . Imagine appropriate vertical arrows that would make the following diagram commute

It follows there is a chain of fields . Taking any lifts of and using that is integral over , it follows that the lift satisfies some polynomial relation with integer coefficients, reducing mod gives a relation for with coefficients in . It follows that the latter residue fields above are finite extensions of a finite field, hence themselves finite fields.

Result: ; more generally in a Dedekind domain iff .

So the upshot is for any prime upstairs there is a unique prime downstairs, and for any prime we can consider the ideal and since we’re in a Dedekind domain so there is some finite set of primes

Ramification index: . The residue class degree: .

Thm: fix a prime downstairs, then .

a prime downstairs ramifies if some , splitting completely means all .

Now thinking of as an vector spaces and looking at left multiplication as an linear map, we can define determinant and trace maps. In particular, for , let denote the linear operator that is multiplication by . Then is and is . These things sends to . (prove this)

Proposition: given by is a nondegenerate symmetric form.

Now define the relative discriminant as the ideal generated by as the matrix for ranges over all bases for over .

Thm: ramifies in if and only if .

Here is the rest of the story. A wonderful fact is that when is Galois, then permutes the primes above any given , in a transitive manner. In this case don’t vary as varies. Let under this action. Naturally given an automorphism preserving and fixing , hence an element of ; this is called the Artin map.

Proposition: is surjective.

A question

Consider and , this is a degree 2 extension and is the splitting field of . The poly has discriminant -4, and the relative discriminant as defined above seems to be -2, in either case they pick out the prime 2, but his seems to split completely , what’s going on?

Actually nothing is going on, , so indeed the prime does ramify.

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- Published:
- September 4, 2009 / 12:54 pm

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- Number Theory (course)

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