If $K=\{a+b\sqrt2\mid a,b\in\mathbb{Q}\}$, find $[K ∶\mathbb{Q}]$ and $[K(\sqrt3) ∶\mathbb{Q}]$.

Question

I'm new to the subject and struggling to understand the steps when finding the degree of a field extension, I've been finding the minimal polynomial and then using the degree of that as the answer, but here, I'm not sure how to do it

Please could somebody talk me through the steps! It's very very much appreciated

Answer 1

$[K ∶\mathbb{Q}]$ is the dimension of the vector space $K$ over the field $\mathbb{Q}.$ I will make you understand via basis. A basis of $K$ over the field $\mathbb{Q}$ is $\{1,\sqrt{2}\}.$ So the dimension is 2. Now $[K (\sqrt{3})∶\mathbb{Q}]=[K(\sqrt{3})∶K].[K∶\mathbb{Q}]$. Can you complete?