# Elliptische Kurve/Y^2 ist X^3+2/2-Torsion/Aufgabe/Lösung

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Die Nullstellen des Polynoms ${\displaystyle {}X^{3}+2}$ sind über ${\displaystyle {}{\mathbb {C} }}$ die komplexen Zahlen

${\displaystyle -{\sqrt[{3}]{2}},-\zeta _{3}{\sqrt[{3}]{2}},-\zeta _{3}^{2}{\sqrt[{3}]{2}},}$

wobei

${\displaystyle {}\zeta _{3}={\frac {-1+{\sqrt {3}}{\mathrm {i} }}{2}}={\frac {-1+{\sqrt {-3}}}{2}}\,}$

eine dritte primitive Einheitswurzel bezeichnet. Es ist

${\displaystyle {}-\zeta _{3}{\sqrt[{3}]{2}}=-{\frac {-1+{\sqrt {-3}}}{2}}{\sqrt[{3}]{2}}={\frac {1-{\sqrt {-3}}}{2}}{\sqrt[{3}]{2}}\,}$

und

${\displaystyle {}-\zeta _{3}^{2}{\sqrt[{3}]{2}}=-{\frac {-1-{\sqrt {-3}}}{2}}{\sqrt[{3}]{2}}={\frac {1+{\sqrt {-3}}}{2}}{\sqrt[{3}]{2}}\,.}$

Nach Fakt ist somit, da ${\displaystyle {}{\sqrt[{3}]{2}}}$ reell und die anderen Nullstellen nicht reell sind,

1. ${\displaystyle {}\operatorname {Tor} _{2}{\left(E(\mathbb {Q} )\right)}=\{{\mathfrak {O}}\}\,,}$
2. ${\displaystyle {}\operatorname {Tor} _{2}{\left(E(\mathbb {Q} [{\sqrt[{3}]{2}}])\right)}=\{{\mathfrak {O}},\,(-{\sqrt[{3}]{2}},0)\}\,,}$
3. ${\displaystyle {}\operatorname {Tor} _{2}{\left(E(\mathbb {Q} [{\sqrt[{3}]{2}},{\sqrt[{2}]{-3}}])\right)}=\{{\mathfrak {O}},\,(-{\sqrt[{3}]{2}},0),\,({\frac {1-{\sqrt {-3}}}{2}}{\sqrt[{3}]{2}},0),\,({\frac {1+{\sqrt {-3}}}{2}}{\sqrt[{3}]{2}},0)\}\,.}$