Kronecker-Weber Theorem

 Let $\tilde{F}$ be a field (called the algebraic closure of $F$) obtained by adjoining all roots of polynomials over $F$. The maximal abelian quotient of $\mathrm{Gal}\left(\tilde{F}/F \right)$ is the Galois group of maximal abelian extension $F'$ of $F$, which is the largest subfield of $\tilde{F}$ whose Galois group is abelian. Setting $F=\mathbb{Q}$, from the Kronecker-Weber theorem, the maximal extension $\mathbb{Q}'$ of $\mathbb{Q}$ is found by adjoining $\mathbb{Q}$ to all roots of unity. Let $\zeta _{N}$ be a fixed primitive $N$th root of unity and $\mathbb{Q}\left(\zeta _{N} \right)$ be the $N$th cyclotomic field. Then, there exists an isomorphism 

$$\left(\mathbb{Z}/N \right)^{\times}\cong \mathrm{Gal}\left(\mathbb{Q}\left(\zeta _{N}\right)/\mathbb{Q} \right)\;.$$

The Kronecker-Weber theorem is then the statement that every finite abelian extension of $\mathbb{Q}$ is contained in a cyclotomic field. This has some interesting properties, particularly those linking to p-adic numbers (which caught my attention thanks to Optimized Fuzzball discussing p-adic AdS/CFT), which I will write about in a later post. I am also preparing a Lichess study on Ruy-Lopez, which I will link sometime soon.