# Structure of the group $(\mathbb{F}_{2}[x]/(Q ^ { e }))^{*}$, where $Q$ ie an irreducible polynomial over $\mathbb{F}_{2}$

Let $Q$ be an irreducible polynomial over $\mathbb{F}_{2}$, can we find a decomposition of the group $(\mathbb{F}_{2}[x]/(Q ^ { e }))^{*}$ into a direct product of cyclic groups ?.

We know (from $P\in \mathbb{F}_{2}[x]$ for which $(\mathbb{F}_{2}[x]/(P))^{*}$ is a cyclic group) that $(\mathbb{F}_{2}[x]/(Q ^{e}))^{*}$ is not cyclic unless $e=1$ or $\deg(Q) = 1$ and $e \leq3$.

If no analytical decomposition can be found is there any algorithm to find such one ? In that case do we have a bound on the number of cyclic groups in the decomposition ?

• Isn't it obvious that an algorithm exists? – Will Sawin Jun 19 '17 at 21:48
• @WillSawin It's not obvious to me, can you explain how ? – ayman Jun 19 '17 at 23:16