Degrees of extensions by $\alpha_1,\dots,\alpha_p$ of $\mathbb Q$ and $ L$ where $ \mathbb Q \leq L$

Question

Suppose that we have a finite extension $\mathbb Q \leq L$ and that the prime $p\nmid |L:\mathbb Q|$. Suppose also that $\alpha_1, \dots, \alpha_p \notin \mathbb Q$

Is it true that $|L(\alpha_1, \dots, \alpha_p):L| \leq |\mathbb Q(\alpha_1,\dots,\alpha_p): \mathbb Q| $?

If not could someone please provide me with a counter example? I've checked a few examples and I think this has held up so far.

Answer 1

If $F\leq K$ and $F\leq L$, $$\color{red}{[L:F] \geq [LK:K]} \iff [LK:L] \leq [K:F]$$ To see why, just note the diagram below, where arrows represent inclusion. $\require{AMScd}$\begin{CD}K @>>> LK \\ @AAA @AAA \\ F @>>> L \end{CD}

The inequality in red is true because, by second isomorphism theorem: $$[LK:K] = [L:L\cap K] \leq [L:F]$$

Taking $F=\mathbb{Q}, K=\mathbb{Q}(a_1,\cdots,a_n)$ immediately answer your question.