Dimensions in field extensions

Question

How would I be able to determine $[\Bbb Q(\sqrt{42}, \sqrt{-42}):\Bbb Q]$? So far, I think the dimension might be four as the root equation could be $(x^2 + 42)(x^2 - 42)$.

Answer 1

The degree is indeed $4$. Note that $$[\mathbb Q(\sqrt{42},\sqrt{-42}):\mathbb Q] = [\mathbb Q(\sqrt{42},\sqrt{-42}):\mathbb Q(\sqrt{42})]\cdot [\mathbb Q(\sqrt{42}):\mathbb Q]$$ and both the extensions on the right have degree at most $2$ since they are obtained by adjoining a square root, and degree more than $1$ since $\sqrt{42}\notin \mathbb Q$ and $\sqrt{-42}\notin \mathbb Q(\sqrt{42})$ since $\sqrt{-42}$ is imaginary.