In this talk we will not prove Fermat's Last Theorem, which states that for $n>2$ there are no positive integer solutions to the equation $a^n + b^n = c^n$. There are many ways to not prove this theorem, and our chosen method will be to explain a theorem of Ribet exhibiting 691-torsion in the class group of the 691st cyclotomic field, and then explain why this ruins an otherwise perfectly nice strategy for proving the titular theorem. In the course of failing to produce a proof, we will encounter class field theory, modular forms, abelian varieties, and Galois representations. I will not presuppose knowledge of these concepts and this talk should be accessible to anyone familiar with number fields.