Why extension problem? Wiki https://en.wikipedia.org/wiki/Solvable_group Why studying group extensions?

+) Answer questions regarding $\mathbb{Q}$ by translating into questions about its completions: Hasse-Minkowski. —-

Field extensions:

+) Cyclotomic extension: $K/F$ where $K$ is splitting field of $X^n-1$ over $F$. +) Cyclic extensions: finite Galois extension $K/F$ such that $\text{Gal}(K/F)$ is cyclic. +) Kummer theory: study abelian extensions containing “enough” roots of unity. +) Local class field theory: study abelian extensions of local number fields.