This is a problem given in the exercise of my text book(Chapter: Algebraic Extension of Fields) But I don't understand the question properly.
Let $D$ be an integral domain and $F$ be a field in $D$ such that $[D:F]<∞$. Prove, $D$ is a field.
If $D$ is not a field then what does $[D:F]$ denote? I know if $E$ is an extension field over a field $F$, then $[E:F]$ denotes the order of $E$ over $F$.
Can anybody clear my queries? Thanks for help in advance.
For a field extension $F\subset E$, $[E:F]$ denotes the dimension of $E$ as an $F$ vector space. I think it is also written as $|E:F|$ sometimes as well.
From page 1 of your book:
I think it is reasonable to assume that it means the same thing if $E$ is replaced with any $F$ algebra.