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Salix alba (talk | contribs) |
→Galois theory of the modular curve: I fixed a little thing here in the field, but there are much deeper issues in this section of the article. For one, the way it is worded suggests that the modular curve X_0(p) is Galois over X(1) (it isn't!) |
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The Galois theory of the modular curve was investigated by [[Erich Hecke]]. Considered as a polynomial in x with coefficients in {{math|'''Z'''[''y'']}}, the modular equation {{math|Φ<sub>0</sub>(''n'')}} is a polynomial of degree {{math|''ψ''(''n'')}} in {{mvar|x}}, whose roots generate a [[Galois extension]] of {{math|'''Q'''(''y'')}}. In the case of {{math|''X''<sub>0</sub>(''p'')}} with {{mvar|p}} prime, where the [[Characteristic (algebra)|characteristic]] of the field is not {{mvar|p}}, the [[Galois group]] of {{math|'''Q'''(''x'', ''y'')/'''Q'''(''y'')}} is {{math|PGL(2, ''p'')}}, the [[projective linear group|projective general linear group]] of [[Möbius transformation|linear fractional transformations]] of the [[projective line]] of the field of {{mvar|p}} elements, which has {{math|''p'' + 1}} points, the degree of {{math|''X''<sub>0</sub>(''p'')}}.
This extension contains an algebraic extension {{math|''F''/'''Q'''}} where if <math>p^* = (-1)^{(p-1)/2}p</math> in the notation of [[Quadratic_reciprocity#Gauss.27s_version_in_Legendre_symbols|Gauss]] then:
:<math>F = \mathbf{Q}\left(\sqrt{
If we extend the field of constants to be {{mvar|F}}, we now have an extension with Galois group {{math|PSL(2, ''p'')}}, the [[projective linear group|projective special linear group]] of the field with {{mvar|p}} elements, which is a finite simple group. By specializing {{mvar|y}} to a specific field element, we can, outside of a thin set, obtain an infinity of examples of fields with Galois group {{math|PSL(2, ''p'')}} over {{mvar|F}}, and {{math|PGL(2, ''p'')}} over {{math|'''Q'''}}.
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