We take an historical approach to abstract algebra: studying approaches to prove one of the most famous theorems in mathematics, Fermat's Last Theorem (FLT). This is somewhat of a misnomer, because Fermat did not prove this theorem; he merely made a private note in the margin of one of his books claiming to have a proof. After his death, when the margin note, but no proof, was discovered, mathematicians took on the challenge of finding a proof. The "theorem" turned out to be more subtle than anyone expected, and, while a proof eluded mathematicians for centuries, their prodigious efforts drove the creation of abstract algebra. We will start our study in the concrete setting of the integers, then move to more exotic mathematical structures like the Gaussian integers and integers modulo n, which motivate the more abstract notion of a commutative ring. We continue with field theory to study Abel and Gauss' work towards proving FLT, and end with a brief overview of the way in which the theorem was finally proven, by Wiles and Taylor in the 1990s.
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