Sophie Germain: mathématicienne extraordinaire - Scripps College

And Beyond 68
In 1847, Kummer published a paper which showed that unique factorization
does indeed fail over certain number rings. He then presented a
new type of element, ideals, which he proved do have unique factorization.
This new approach did not lead to a final solution of Fermat’s Last
Theorem, but it created a new branch of mathematics, the field of Algebraic
Number Theory. [6]
Wiles’ Proof (The bare bones).
In 1986, it was found that the Shimura-
Taniyama conjecture on elliptic curves was related to Fermat’s Last Theorem.
Most elliptic curves are give by equations of the form:
E : y 2 = x 3 + ax 2 + bx + c,
where a, b, and c ∈ Q. Elliptic curves have the structure of an abelian group.
That is, given any rational point on the curve, it is possible to find another
rational point, should such points exist. In addition, given a prime p under
certain conditions (i.e., not a bad prime), one can reduce certain elliptic
curves modulo p which produce certain patterns of rational numbers. [23]
In 1993 Andrew Wiles announced his proof of Fermat’s Last Theorem
as a corollary to his proof that semistable elliptic curves exhibit Modularity
Patterns. A result is that certain elliptic curves, in particular, the Frey curve:
E A,B : y 2 = x(x + A p )(x − B p )
must be modular. Ken Ribet had earlier proved that the Frey curve cannot
be modular. A curve cannot be both modular and non-modular, so