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!


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 $2$.

| cite | improve this answer | |

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.