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== Galois theory of the modular curve ==
The Galois theory of the modular curve was investigated by [[Erich Hecke]]. Considered as a polynomial in x with coefficients in '''Z'''[y], the modular equation Φ<sub>0</sub>(n) is a polynomial of degree ψ(n) in x, whose roots generate a [[Galois extension]] of '''Q'''(y). In the case of X<sub>0</sub>(p) with p prime, where the [[Characteristic (algebra)|characteristic]] of the field is not p, the [[Galois group]] of
:'''Q'''(x, y)/'''Q'''(y)
This extension contains an algebraic extension <math>F = \mathbf {Q}(\sqrt{(-1)^\frac{p-1}{2}p})</math> of '''Q'''. If we extend the field of constants to be ''F'', we now have an extention with Galois group PSL<sub>2</sub>(p), the [[projective linear group|projective special linear group]] of the field with p elements, which is a finite simple group. By specializing y to a specific field element, we can, outside of a thin set, obtain an infinity of examples of fields with Galois group PSL<sub>2</sub>(p) over ''F'', and PGL<sub>2</sub>(p) over '''Q'''. ▼
is PGL<sub>2</sub>(p), the [[projective linear group|projective general linear group]] of [[Möbius transformations|linear fractional transformations]] of the [[projective line]] of the field of p elements, which has p+1 points, the degree of X<sub>0</sub>(p).
When n is not a prime, the Galois groups can be analyzed in terms of the factors of n as a wreath product.▼
This extension contains an algebraic extension
:<math>F = \mathbf {Q}(\sqrt{(-1)^\frac{p-1}{2}p})</math>
▲
▲When ''n'' is not a prime, the Galois groups can be analyzed in terms of the factors of ''n'' as a [[wreath product]].
== See also ==
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