I was asked to show that the group of invertible elements of the ring $\Bbb{F}_p[x]/\langle x^{p+1}\rangle$ is isomorphic to the abelian group $\Bbb{Z}_{p-1}\times\Bbb{Z}_{p^2}\times\left(\Bbb{Z}_p\right)^{p-2}$ and I have no idea to begin the solution. Can any body provide me a way to solve this problem. Thank you in advance for your support.
Here $p$ is a prime.
In general, no matter what the base field $k$, an element of $k[x]/\langle x^n\rangle$ is uniquely writable as $\lambda u$ where $\lambda\in k^\times$, a group of order $p-1$ in our case, and $u\in1+\langle x\rangle$, the group of principal units in this local ring. When you analyse the structure of $1+\mathfrak m=1+\langle x\rangle$, where $\mathfrak m$ is the maximal ideal of this ring, you will have the whole story.