Square-class group of the 2-adic numbers (${\mathbb{Q}}_2^{\times }/({\mathbb{Q}}_2^{\times }{)}^2$)

An odd $2$-adic unit is a square exactly when it is congruent to $1$ modulo $8$. Valuation parity and the four odd residue classes modulo $8$ therefore givewith generators represented by $-1$, $2$, and $5$.