Field extension-degree

Question

I have the following question...

$K\leq E$ a field extension.

When we have that $$[E:K]=1$$ do we conclude that $K=E$??

Or must also something else be satisfied so that $K=E$ ??

Answer 1

Yes, you can conclude that $K=E$. In fact the degree of an extension $K\subseteq E$ is the dimension of $E$ as a $K$-vector space. Hence $E=K$.

Answer 2

It means $\;\deg_KE=1\;$ , so

$$\deg_KK=\deg_KE=1\;\;\text{and also}\;\;K\le E$$

from both conditions above it follows from linear algebra that $\;E=K\;$ , so it is always true.