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Subfield and field extension

Definition

If $F \subset E$ are fields (with the same operations), we say $F$ is a subfield of $E$ and $E$ is a field extension of $F$.

Criterion

Subfield criterion: a subset $S \subseteq F$ is a subfield if $0,1 \in S$ and for all $a, b \in S$ we have $a-b, ab \in S$ and if $a \ne 0$ then $a^{-1} \in S$

Properties

The intersection of all subfields $S \subseteq F$ is a field, called the prime subfield of $F$.

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 $[K:F] = [K:E][E:F]$

Examples

Suppose $F$ is not algebraically closed and let $f(x) \in F[x]$ be an irreducible polynomial. Then (TODO)

all these pages adapted with probably insufficient credit from my university's lecture notes