I don't quite understand how to find the degree of a field extension. First, what does the notation [R:K] mean exactly?
If I had, for example, to find the degree of $\mathbb Q (\sqrt7)$ over $\mathbb Q$, how would I go about it? And how would it be different from, say, $\mathbb C (\sqrt7)$ over $\mathbb C$?
Would it involve finding the minimal polynomial? In the case of $\mathbb Q (\sqrt7)$, I find the minimal polynomial to be $x^2-7$ which is of degree 2, so would this be the value of $[\mathbb Q (\sqrt7):\mathbb Q]$?
In $\mathbb C $, this polynomial is reducible, so I assume somehow it would need to be reduced to find the minimal polynomial. (I am guessing since $x^2-7=(x^2+1)-8=i-8$ this is the polynomial in $\mathbb C$) Would the degree of this be the degree of$[\mathbb C (\sqrt7):\mathbb C]$?