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Use of Lagrange theorem (group theory) to prove Fermat's small theorem?

Use of fixed point methods (and completeness) to prove existence of solutions to differential equations?

Use of Fields theory to prove the impossibility of the trisection of the angle?

Use of group theory to prove insolvability of 5th degree equation?

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Use of Lagrange theorem (group theory) to prove Fermat's small theorem?

Use of fixed point methods (and completeness) to prove existence of solutions to differential equations?