Number Theory Lecture 5

 

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

Excerpts

An aboslute value on a field F is a fucntion |-|\colon F \to \mathbb{R}_+ s.t

  1. |x| = 0 \Longleftrightarrow x = 0
  2. \forall x,y \in F, |xy| = |x|\cdot|y|
  3. \forall x,y \in F, |x+y| \le |x| + |y|

With these properties it can be checked the norm of 1 is 1 and |1/x| = 1/|x| and you can define a metric via d(x,y) = |x - y|.

If you have a field with a discrete valuation v then you can get a absolute value essentially by turning the valuation on its head: c \in (0,1), then define |x|_{v,c} = c^{v(x)}.

An aboslute value is non archimedean if |x+y| \le \max (|x|,|y|), \forall x,y.  If this fails then |-| is called archimedean.

Facts on this stuff

  1. Any F with an archimedean absolute value is isomorphic to a subfield of \mathbb{C}.
  2. For a non archimedean absolute values, \bar B_r(a) = \{|x - a|\le r\} is open and all triangles are isoceles.
  3. If F, |-| is non archimedean, then the induced topology makes F totally disconnected.
  4. With F, |-| as above, R = \{x| |x| \le 1\} is a ring with maximal ideal M = \{|x|<1\}.

Back to number fields

The basic story we’ve been following consists at looks at extension of number fields F \to E, and its ring of integers O_F \to O_E.  In the case of Galois extension there’s a nice story about a prime downstairs p \subset O_F and the primes lying over it P_i \subset O_E.  This lead to looking at the Galois group Gal(E/F) which permutes the P_i and by looking at the stabilizer we get a map \eta \colon Stab(P_i) \to Gal(F_P/F_p) which gives us a canonical Frobenius conjugacy class when the prime p doesn’t ramify, and we can test for this with the descriminant.

Now taking a \in F^* we can look at the factional ideal this generates and this factor into a product of prime ideals to various powers (a) = \prod_i p_i^{n_i}, so naturally a prime gives a valuation.  Taking c = 1/|F_p| \in (0,1) provides an absolute value (non archimedean) on F^*.  Since everything gets turned on its head, its not hard to check that the ring R,m associated to an absolute value in this case is just O_{F,p}.

Here is the upshot.  For a number field F/\mathbb{Q}, taking a prime in O_F gives a non archimedean absolute value on F.  An embedding \sigma \colon F \to \mathbb{C} given an archimedean absolute value.

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

A place on a number field F is an equivalence class of a non-trivial absolute value.  For example, F = \mathbb{Q}[x]/(x^2 + 1) has two embeddings and one place.  Archimedean places are called infinite places, non archimedean are finite places.

Thm: Let F a number field and x \in F^*, let v index places on F.

\prod_{v \infty} |x|_v^{e(v)} \cdot \prod_{v \mbox{ finite}} |x|_v = 1

Let’s verify this for F = \mathbb{Q}.  For a nonzero rational, r/s = \prod p_i^n_i \prod q_j^{-n_j}.  Let v be the place corresponding p_i, then |x|_v = (1/p_i)^{n_i}, in the case of negative exponent, we get |x|_v = (1/q_j)^{-n_j} = q_j^{n_j}.  Of course there is only one infinite place that is just the usual absolute value on \mathbb{C}.  

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