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A [[global field]] is either a [[Field extension|finite algebraic extension]] of <math>\Q</math>, or a [[Field extension|finite algebraic extension]] of the filed of rational functions over a finite field. In the first case the field is called a number field, and in the latter case the field is called a function field.

The Chebotarev density theorem can be generalized to global fields.
Let $K$ be a global field. The ring of integers of <math>K</math> is denoted by <math>\mathcal{O}_K</math>. In the case <math>K</math> is a number field, <math>\mathcal{O}_K</math> is the integral closure of <math>\Z</math> in <math>K</math>. In the case that <math>K</math> is a function field, <math>\mathcal{O}_K</math> is the integral closure of '''F'''<sub>''p''</sub> in <math>K</math>.

\textbf{The Artin Symbol}
Let <math>L/K</math> be a Galois extension of <math>K</math>. Let <math>\mathfrak{p}</math> be an ideal in <math>\mathcal{O}_K</math> and let <math>\mathfrak{B}</math> be an ideal in <math>\mathcal{O}_L</math>. The decomposition group <math>D_{\mathfrak{B}}</math> is a subgroup of <math>Gal(L/K)</math> which is defined by

:<math>\{|D_{\mathfrak{B}}=\{\sigma \in G | \sigma(\mathfrak{B})=\mathfrak{B}\}</math>


The inertia group <math>I_\mathfrak{B}</math> is a subgroup of <math>D_{\mathfrak{B}}</math> which is defined by

:<math>I_{\mathfrak{B}}=\{\sigma \in G | \sigma(x) \in x + \mathfrak{B} \mbox{ for every x } \in O_L\}</math>

Let <math>K_{\mathfrak{p}}</math> denote the [[Localization of a ring|localization]] of <math>O_K</math> by <math>\mathfrak{p}</math>. The residue field of <math>K_{\mathfrak{p}}</math> is a finite field. Denote it by <math>\overline{K}_{\mathfrak{p}}</math>, and let <math>N_{\mathfrak{p}}=|\overline{K}_{\mathfrak{p}}|</math>.

There is a canonical homomorphism from <math>D_{\mathfrak{B}}</math> to <math>Aut(\bar{L}/\bar{K})</math> which is given by


:<math>\overline{\sigma} (\overline{x})= \overline{\sigma (x)}</math>


having <math>I_{\mathfrak{B}}</math> as its kernel.
If <math>\mathfrak{p}</math> is un-ramified in <math>\mathcal{O}_L</math>, <math>\bar{L}/\bar{K}</math> is Galios and <math>I_{\mathfrak{B}}=\{1\}</math> and the homomorphism defined above is in fact an isomorphism.

In <math>Gal(\bar{L}/\bar{k})</math> there is a distinctive element, which is the frobinius automorphism <math>Frob \in Gal(\bar{L}/\bar{k})</math>. The element of $Gal(L/K)$ which maps to <math>Frob</math> by the isomorphism described above is denoted by <math>\left[\frac{L/K}{\mathfrak{B}}\right]</math>. It satisfies

:<math>\left[\frac{L/K}{\mathfrak{B}}\right] x \equiv x^{N_{\mathfrak{p}}}(x) \mod \mathfrak{B}</math>

if <math>\sigma \in Gal(L/K)</math> then <math>\left[\frac{L/K}{\mathfrak{B}}\right] = \sigma \left[\frac{L/K}{\mathfrak{B}}\right] \sigma^{-1}</math>. Hence

:<math>\{\left[\frac{L/K}{\mathfrak{B}}\right] \in Gal(L/K) | {\mathfrak{B}} \mbox{ lies over } \mathfrak{p}\}</math>

is a conjugacy class in <math>Gal(L/K)</math>. This conjugacy class is the Artin Symbol and is denoted by <math>\left(\frac{L/K}{\mathfrak{p}}\right)</math>. The Artin Symbol is defined for every <math>\mathfrak{p}</math> which is unramified in <math>O_L</math>.

\textbf{Derichlet density}

Let <math>P(K)</math> denote the set of prime ideals of <math>\mathcal{O}_K</math>. For a subset <math>A</math> of <math>\mathcal{O}_K</math> The Dirichlet density is defined by

:<math>\delta(A)=\lim_{s \to 1^{+}} \frac{\sum_{\mathfrak{p} \in A}(N_{\mathfrak{p}})^{-s}}{\sum_{\mathfrak{p} \in P(K)}(N_{\mathfrak{p}})^{-s}}</math>


The Chebotarev density theorem for global fields is

Let <math>L/K</math> be a finite Galois extension of global fields and let <math>C</math> be a conjugacy class in <math>Gal(L/K)</math>. Then the Dirichlet density of <math>\{\mathfrak{p} \in P(K)|\left(\frac{L/K}{\mathfrak{p}}\right) = C\}</math> exists and is equal to <math>\frac{|C|}{[L:K]}</math>

Revision as of 16:28, 30 July 2014

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