Let $k = F_p$, the finite field of $p$ elements and $F = k(t, u)$ where $t$ and $u$ are independent variables. Let $α$ and $β$ be roots of $x^p − t$ and $x^p − u$ respectively. Let $L = F(α, β)$. Consider the intermediate fields $F ⊂ F(α+λβ) ⊂ L$ as $λ$ varies over all elements of $F$. I want to show there are infinitely many intermediate fields between F and L in the following way.
So let's say $λ ≠ µ$ are two elements of $F$ such that $F(α + λβ) = F(α + µβ)$, how to show that $α, β ∈ F(α + λβ)$? If I can do this then I can conclude that $F(α + λβ) = F(α, β)$ which is not possible right? :O