2-adic unit ( $Z_{2}^{\times}$ )

A 2-adic unit is an odd element of

Z_{2}

. It is a square in

Z_{2}^{\times}

exactly when it is congruent to

1

8

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

An odd

2

-adic unit is a square exactly when it is congruent to

1

8

. Valuation parity and the four odd residue classes modulo

8

therefore give

Q_{2}^{\times} / (Q_{2}^{\times})^{2} ≅ (Z /2 Z)^{3},

with generators represented by

- 1

2

, and

5

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