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I am very new to field extensions, so I am very sorry for this basic question.

I want to determine the degree of the field extension $$[\mathbb{Q}(\sqrt{3}, i\sqrt{2}):\mathbb{Q}(\sqrt{3})]$$

So my problem now is that I dont know what the minimal polynomial could look like. I have only dealt with field extensions of a single element over a "normal" field such as $\mathbb{Q}$.

Help is very much appreciated!

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The extension is generated by a single element $i\sqrt{2}$, so the extension degree is just the degree of the minimal polynomial of $i\sqrt{2}$ over $\mathbb{Q}(\sqrt{3})$. First of all note that $i\sqrt{2}\notin\mathbb{Q}(\sqrt{3})$ because $\mathbb{Q}(\sqrt{3})$ is a field which contains only real elements. So the extension degree can't be $1$, it is at least $2$. Also, it is easy to see that $i\sqrt{2}$ is the root of the polynomial $x^2+2\in\mathbb{Q}(\sqrt{3})[x]$, so the extension degree is also at most $2$. Hence it is exactly $2$.

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