Catalan's Conjecture - A learning exercise for the bored mind : 2

Catalan's Conjecture - A learning exercise for the bored mind : 2.md

Before we move to Mihailescu's proof, let us cover some more mathematical concepts (so that we feel superior to the people around us, just kidding 😄),

There is an interesting theorem that Mihailescu uses in his proof, called the Stickelberger's theorem,

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Stickelberger's theorem:

This is a result of algebraic number theory, which gives more information about Galois Module structure of class groups of Cyclotomic Fields. This theorem consists of Stickelberger's element and Stickelberger's ideal. I will now state the complete definition of the theorem and then visit its corners as we move along,

Let $K_m$ denote the $m$-th cyclotomic field${[1]}$ . It is a Galois extension of $\mathbb{Q}$ with Galois Group $G_m$ isomorphic to the multiplicative group of integers modulo $m$ $(\mathbb{Z}/m\mathbb{Z})^{\times}$.

The Stickelberger