Ring theory Algebra

Question

Let F be a field of $8$ elements and A=set of all x belongs to F such that $x^7=1$ and $x^k \neq 1$ for all $k < 7$. then the number of elements in A is

Answer 1

Note that $0 \in F$ is not a unit. Therefore, we are really interested in looking at $F^\times = F$ \ $\{0\}$, the set of units in $F$. The important thing to recognize is that $F^\times$ forms a group under multiplication.

Note that $F^\times$ forms a group that is, in particular, of order $7$, which is prime. All groups of prime order are cyclic $\Longrightarrow F^\times \cong \mathbb{Z}_7$. Can you now apply Lagrange's theorem to finish up?

Answer 2

If $F$ is a field then every finite subgroup $G \subset F^{\times}$ is cyclic. In particular in our case $ F^{\times}$ is cyclic of order $7$, and in a cyclic group of order $7$ there are $6$ elements of order $7$, i.e. all the elements except the unit