A problem in Galois rings

Question

Let $\xi\in GR(4^{m})$ and $\xi$ is of order $2^{m}-1$. Let $a=l-k$ for l and k are distinct in the range $[0,2^{m}-2]$, where $m\geq 2$. Why $2\xi^{a}=0$ is a contradiction? Could you help me to explain the reasons?

Answer 1

Since $\xi$ is invertible, so is $\xi^a$. Multiplication of the equation $2\xi^a = 0$ with the inverse of $\xi^a$ yields the contradiction $2 = 0$ (note that the characteristic of your base ring is $4$).