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subfield

Definition

A subfield is a field contained within another field (sharing the identities and operations, of course).

Properties

If F is a subfield of E, we call E a field extension of F, and we can consider E as a vector space over F, denoted $E/F$. We call its dimension the degree of the field extension, denoted $[E:F] = \dim_F(E) = \dim(E/F)$

Tower law: If $F \subset E \subset K$ are fields, then [K:F] is finite if & only if [K:E] and [E:F] are finite. In this case we have $[K:F] = [K:E][E:F]$