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+I am such a skilled editor!
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-In the original Sherlock Holmes (1895), Evil Professor Moriarty stated that when he retired he was going to devote his time studying Abstract Mathematics.
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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 <math>K</math> 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>.
-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 [[Ramification#In algebraic number theory|unramified]] 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 <math>Gal(L/K)</math> 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>.
-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>
-I am an "anonymous" testing edit. I am another edit. And another. Yet one more.