Problem 8

Let $F$ be a finite field with characteristic $p$, a prime.

1. Explain why $p$ divides $|F|$.

2. Suppose $q$ is another prime dividing $|F|$. Show there exist integers $\lambda$ and $\mu$ such that $(\lambda p+\mu q)x=x$ for all elements $x\in F$.

3. Deduce that the supposition in (ii) leads to a contradiction. Hence explain why $|F|$ must have order $p^{n}$, where $n$ is a positive integer.