Return to Revisions

For a prime $p \neq 2$, the multiplicative group $\mathbb{Q}_p^\times$ is isomorphic to $\mathbb Z \times \mathbb Z_p \times \mathbb Z/(p-1)$, while $\mathbb Q_2^\times$ is isomorphic to $\mathbb Z \times \mathbb Z_2 \times \mathbb Z/2$, cf. Serre, A course in arithmetic, Theorem II.3.2 (p. 17). Since a field isomorphism $\varphi\colon \mathbb Q_p \to \mathbb Q_r$ will preserve the torsion subgroup of the multiplicative subgroup, this shows that $\mathbb Q_p \not\cong \mathbb Q_r$ whenever $\{p,r\} \neq \{2,3\}$. The remaining case is taken care of by the fact that $\mathbb{Q}_p^\times / \mathbb {Q}_p^\times^2 \cong \mathbb Z/2 \times \mathbb/2$ whenever $p \neq 2$, while $\mathbb Q_2^\times / \mathbb Q_2^\times^2 \cong \mathbb Z/2 \times \mathbb Z/2 \times \mathbb Z/2$, cf. loc. cit., p.18.