Def for $\dim_{\mathbb{Q}}(M)$?

Question

I'm working on a proof that $M$ is not projective given $M$ is a $\mathbb{Q}[x]$-module such that $\dim_{\mathbb{Q}}(M) < \infty$. What does $\dim_{\mathbb{Q}}(M)$ mean? I know the the proof has something to do with the $M$ not being projective iff for any module $Q$ we have $Q\bigoplus M$ is not a free module. However I need that definition of $\dim_{\mathbb{Q}}$ to really proceed. Tips for references to that definition or statements of it? I've pretty much exhausted Wikipedia without knowing the exact phrase.

p.s. if it's surprising that I'm working on this problem but don't have the definition I am doing math way beyond me for fun but don't have access to books on college level algebra at home.

edit: I found this question shortly after posting and though it doesn't solve my question it has me wondering if this is the definition I'm looking for. I suspect it might be but they do not use the same notation so it is still an opaque question to me.

Answer 1

Thanks to u/JDZ I believe the answer is as follows.

$\dim_{\mathbb{Q}}(M)$ is the dimension of $M$ as a vector space over $\mathbb{Q}$.

Also from another source it seems that $\dim_{\mathbb{Q}}(M)$ is the size of the "basis vector" of elements in $M$ that elements of $\mathbb{Q}$ are multiplied by to get all of $M$ in the summands.