Find the degree of a finite field extension

Question

What is the degree of the extension $[\mathbb{Q}(\sqrt{2},\sqrt{3},\sqrt{6}):\mathbb{Q}]$? Is it true that this extension is equal to $\mathbb{Q}(\sqrt{2},\sqrt{3})$?

Answer 1

We have that $\mathbb{Q}[\sqrt[]{2}, \sqrt[]{3}]$ contains $\sqrt[]{6}$ given that its elements must be closed under multiplication and so the product $(0+\sqrt[]{2})(0+\sqrt[]{3})=\sqrt[]{6}$ must be contained in the set. So extending it to $\sqrt[]{6}$ is redundant.

To determine the degree of the extension consider what elements need to be included in the basis so that when our scalars consist of elements from $\mathbb{Q}$, we can get all of the rationals, and any multiple of the elements of the extension so that it is closed under mutliplication and addition.