order of $\mathbb{Z}_3^{\times}/(\mathbb{Z}_3^{\times})^3$

Question

Based on this: What is the group structure of 3-adic group of the cubes of units?, we know that $(\mathbb{Z}_3^{\times})^3$ are the elements of $\mathbb{Z}_3$ that are congruent to $(+/-)1 \mod 9$. How can we know the order of $\mathbb{Z}_3^{\times}/(\mathbb{Z}_3^{\times})^3$? Thanks in advance!

Answer 1

In general, one has the explicit isomorphism $ \mathbf Z_p^{\times} \cong \mathbf Z/(p-1) \mathbf Z \oplus \mathbf Z_p $ for odd $ p $. The subgroup $ (\mathbf Z_3^{\times})^3 $ corresponds under this isomorphism to the subgroup $ (3 \mathbf Z/2\mathbf Z) \oplus 3 \mathbf Z_3 = \mathbf Z/2\mathbf Z \oplus 3 \mathbf Z_3 $, so taking quotients yields $ (\mathbf Z_3^{\times})/(\mathbf Z_3^{\times})^3 \cong \mathbf Z_3/3 \mathbf Z_3 \cong \mathbf Z/3 \mathbf Z $.