Example 2

\* Let us take $K = \mathbf{Q}$ and $n = 2$. Then $\mathbf{Q}^*/\mathbf{Q}^{*2}$ is a vector $\mathbf{F}_2$-space having as a basis the union of $\{-1\}$ and the set of all prime numbers. The largest abelian extension of exponent 2 of Q contained in $\mathbf{C}$ is thus the subfield $\mathbf{Q}(i, \sqrt{2}, \sqrt{3}, \sqrt{5}, \ldots)$ of $\mathbf{C}$. Its Galois group consists of all the automorphisms obtained by multiplying each of the elements $i,\ \sqrt{2},\ \sqrt{3},\ \sqrt{5}$ etc. by $\pm 1$. \*