We say that a field $K$ is an extension of $F$ if $F$ is a subfield of $K$ .

We usually denote $K$ being an extension of $F$ by $F\subset K$ , $F\le K$ , $K/F$ or

If $K$ is an extension of $F$ , we can regard $K$ as a vector space over $F$ . The dimension of this space (which could possibly be infinite) is denoted $[K:F]$ , and called the degree of the extension.¹

One of the classic theorems on extensions states that if $F\subset K\subset L$ , then $$[L:F]=[L:K][K:F]$$ (in other words, degrees are multiplicative in towers).

Footnotes

... extension.¹

The term ``degree'' reflects the fact that, in the more general setting of Dedekind domains and scheme-theoretic algebraic curves, the degree of an extension of function fields equals the algebraic degree of the polynomial defining the projection map of the underlying curves.