While working through my book I've run into a question where I'm not too sure what is being asked of me/how to start thinking about it. It states:
Suppose $E$ is an extension field of $\mathbb{Z}_7$ and $d \in E -\mathbb{Z}_7$. Find $\deg(d/\mathbb{Z_7})$ if
a) $d^{5}=2$
...
and then they list a couple more cases for me to figure out. How am I supposed to figure these types of things out? I feel like this is going to be a one trick wonder but I've been looking at it and been at a loss.
Check first if $2$ is already a $5$th power; for example $4$ is not only a square it is a $5$th power too: $2^5=32\equiv 4\pmod 7$. And $3^5=243\equiv5\pmod 7$. Their product and inverses are also fifth powers which are $6,2,3$ respectively, and $1$ is always a power. So every number is a 5th power in this field.