I was looking over an old exam paper and I came across a question which confused me, it says:
Let $E = GF(7)(t)$ be the ﬁeld of rational functions in the indeterminate $t$ over $GF(7)$. Deﬁne $σ,τ ∈ Aut(E)$ by $σ(t) = 2t$ and $τ(t) = 1/ t.$ Set $G = \langle σ,τ\rangle$ and $F = Fix(G)$.
Find the minimal polynomial of $t$ over $F$
Well I know that $\sigma(t)$ will be a root to some min. polynomial which would lead me to believe that $2t,4t$ and $t$ were roots but then we'd get the min polynomial $x^3-7x^2t+7xt^2-t^3=x^3-t^3$
But then our minimal polynomial has two indeterminates which seems meaningless to me , could anyone please shed some light on what I'm misunderstanding ?