What is the group structure of 3-adic group of the cubes of units?

From what I understand about the group of units of the 3-adic integers, it consists all $a$ that can be expressed as a = $\sum_{i=0}^\infty a_i 3^i$ where $a_0 \neq 0$. How can we recognize which 3-adic integers are the cubes of units? I was also told that cubes of units of 3-adic integers form a group. What does this group look like?

Solution 1:

If $a\in\Bbb Z_3$ with $a\equiv1\pmod{27}$ then the equation $x^3=a$ is soluble in $\Bbb Z_3$. This is because of this form of Hensel's lemma: If $|f(x_0)|_p<|f'(x_0)|_p^2$ then $f(x)\equiv 0$ is soluble. Here we take $x_0=1$. So the group of cubes $C$ lies between $U$, the unit group and $U_3$ of the numbers $\equiv1\pmod{3^3}$. To find $C$ exactly, consider the cubes in the quotient group $U/U_3\cong(\Bbb Z/27\Bbb Z)^*$. There are $6$ of these congruent to $\pm 1$, $\pm10$, $\pm19$. So the cubes are exactly those elements in $U$ congruent to $\pm1\pmod 9$.