# Infinitely many intermediate fields between F and L

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

Assume that $$F(\alpha + \lambda\beta) = F(\alpha + \mu\beta)$$ for some $$\lambda, \mu \in F$$. Then we have that $$\alpha + \mu\beta \in F(\alpha + \lambda\beta)$$ and so:

$$(\mu - \lambda)\beta = \alpha + \mu\beta - (\alpha + \lambda\beta) \in F(\alpha + \lambda\beta)$$

This implies that $$\beta \in F(\alpha + \lambda\beta)$$, as $$(\mu - \lambda)$$ is an element of $$F$$. From here it's not hard to conclude that $$\alpha = \alpha + \lambda\beta - \lambda\beta \in F(\alpha + \lambda\beta)$$. Therefore as you've noted we get that $$F(\alpha + \lambda\beta) = F(\alpha, \beta)$$.

To see that this equality is impossible, note that:

$$(\alpha + \lambda\beta)^p = \alpha^p + \lambda^p\beta^p = t + \lambda^pu \in F$$

This means that $$[F(\alpha + \lambda\beta):F] = p$$, while $$[F(\alpha,\beta):F]=p^2$$, which is a contadiction.

• Thank you! Very clear answer, I'm needing help with another qns on galois theory too: math.stackexchange.com/questions/2944604/… Oct 6, 2018 at 16:00
• Follow-up question: Are all the intermediate fields necessarily of this form $F(\alpha + \lambda\beta)$? I guess not, but I'm wondering specifically in the case $p = 2$, if there is a way to characterize all the intermediate fields. Jun 15, 2020 at 23:54