Determine the isomorphism class of group of units of $\mathbb Z_5[i]$.

Question

Unfortunately,I've no idea of dealing with this problem.The term isomorphism class leads me to think of fundamental theorem of finitely generated abelian groups but i think i'm going in wrong direction.

Please give me some hint?suggestions?Anything?

Answer 1

Since $(2+i)(2-i)=5$ in $\mathbb Z[i]$, with the two factors relative prime there, and $\mathbb Z[i]$ is a PID, we can conclude that $$(\mathbb Z[i])/\langle 5\rangle = \mathbb Z[i]/\langle 2+i\rangle \times \mathbb Z[i]/\langle 2-i\rangle.$$

But, by conjugation, $\mathbb Z[i]/\langle2+i\rangle\cong \mathbb Z[i]/\langle 2-i\rangle.$ And the units of the product of two rings is the product of the units of each ring.

So you just need to know the group of units of $\mathbb Z[i]/\langle 2+i\rangle$, and that $\mathbb Z[i]/\langle 5\rangle \cong \mathbb Z_5[i]$.

Show that $\mathbb Z[i]/\langle 2+i\rangle\cong \mathbb Z_5$.

Answer 2

$\Bbb Z_5[i] \cong \Bbb Z_5[x]/\langle x^2+1 \rangle = \Bbb Z_5[x]/\langle(x+2)(x+3)\rangle \cong \Bbb Z_5[x]/\langle x+2 \rangle \times \Bbb Z_5[x]/\langle x+3 \rangle$ since $\langle x+2\rangle + \langle x+3 \rangle = \langle 1 \rangle$.

So it is isomorphic to $\Bbb Z_5 \times \Bbb Z_5$.

So its units are isomorphic to $U(5) \times U(5) \cong \Bbb Z_4 \times \Bbb Z_4$.