# Number Theory Lecture 5

This lecture ( NumTh5 ) was given on 9/10/09

### Excerpts

An aboslute value on a field is a fucntion s.t

With these properties it can be checked the norm of 1 is 1 and and you can define a metric via .

If you have a field with a discrete valuation then you can get a absolute value essentially by turning the valuation on its head: , then define .

An aboslute value is non archimedean if . If this fails then is called archimedean.

Facts on this stuff

- Any with an archimedean absolute value is isomorphic to a subfield of .
- For a non archimedean absolute values, is open and all triangles are isoceles.
- If is non archimedean, then the induced topology makes totally disconnected.
- With as above, is a ring with maximal ideal .

### Back to number fields

The basic story we’ve been following consists at looks at extension of number fields , and its ring of integers . In the case of Galois extension there’s a nice story about a prime downstairs and the primes lying over it . This lead to looking at the Galois group which permutes the and by looking at the stabilizer we get a map which gives us a canonical Frobenius conjugacy class when the prime doesn’t ramify, and we can test for this with the descriminant.

Now taking we can look at the factional ideal this generates and this factor into a product of prime ideals to various powers , so naturally a prime gives a valuation. Taking provides an absolute value (non archimedean) on . Since everything gets turned on its head, its not hard to check that the ring associated to an absolute value in this case is just .

Here is the upshot. For a number field , taking a prime in gives a non archimedean absolute value on . An embedding given an archimedean absolute value.

Thm: All non archimedean and archimedean absolute values on arise in this way. Further, two different embedding give rise to the same absolute value exactly when they differ by complex conjugation.

A place on a number field is an equivalence class of a non-trivial absolute value. For example, has two embeddings and one place. Archimedean places are called infinite places, non archimedean are finite places.

Thm: Let a number field and , let index places on .

Let’s verify this for . For a nonzero rational, . Let be the place corresponding , then , in the case of negative exponent, we get . Of course there is only one infinite place that is just the usual absolute value on .

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- Published:
- September 13, 2009 / 4:52 pm

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- Number Theory (course)

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