Field Extensions and Algebraic Elements

Field extensions provide the framework for studying roots of polynomials, constructibility, and the structure of fields beyond the familiar number fields.

Basic Definitions

Definition7.1Field extension

A field extension $K/F$ consists of fields $F \subseteq K$ where $F$ is a subfield of $K$ . The degree $[K:F] = \dim_F K$ is the dimension of $K$ as an $F$ -vector space. The extension is finite if $[K:F] < \infty$ .

Definition7.2Algebraic and transcendental elements

An element $\alpha \in K$ is algebraic over $F$ if it is a root of some nonzero polynomial $f \in F[x]$ . Otherwise, $\alpha$ is transcendental over $F$ . The minimal polynomial $m_\alpha \in F[x]$ is the unique monic irreducible polynomial with $m_\alpha(\alpha) = 0$ . Its degree $\deg(m_\alpha) = [F(\alpha):F]$ .

ExampleAlgebraic and transcendental elements

$\sqrt{2}$ is algebraic over $\mathbb{Q}$ with minimal polynomial $x^2 - 2$ , $[\mathbb{Q}(\sqrt{2}):\mathbb{Q}] = 2$ .
$i$ is algebraic over $\mathbb{R}$ with minimal polynomial $x^2 + 1$ , $[\mathbb{C}:\mathbb{R}] = 2$ .
$\pi$ and $e$ are transcendental over $\mathbb{Q}$ (Lindemann, Hermite).
$\sqrt[3]{2}$ is algebraic over $\mathbb{Q}$ with $[\mathbb{Q}(\sqrt[3]{2}):\mathbb{Q}] = 3$ .

The Tower Law

Theorem7.1Tower law (multiplicativity of degrees)

If $F \subseteq K \subseteq L$ are fields, then $[L:F] = [L:K] \cdot [K:F]$ . In particular, $L/F$ is finite iff both $L/K$ and $K/F$ are finite.

ExampleApplication of the tower law

$[\mathbb{Q}(\sqrt{2}, \sqrt{3}):\mathbb{Q}] = [\mathbb{Q}(\sqrt{2}, \sqrt{3}):\mathbb{Q}(\sqrt{2})] \cdot [\mathbb{Q}(\sqrt{2}):\mathbb{Q}] = 2 \cdot 2 = 4$ .

The second factor is 2 because $\sqrt{3} \notin \mathbb{Q}(\sqrt{2})$ (otherwise $\sqrt{3} = a + b\sqrt{2}$ , squaring: $3 = a^2 + 2b^2 + 2ab\sqrt{2}$ , forcing $ab = 0$ , contradiction).

Algebraic Extensions

Theorem7.2Characterization of algebraic extensions

A field extension $K/F$ is algebraic (every element is algebraic over $F$ ) if and only if every finitely generated sub-extension is finite. Every finite extension is algebraic, but the converse is false (e.g., $\overline{\mathbb{Q}}/\mathbb{Q}$ ).

RemarkThe algebraic closure

The algebraic closure $\overline{F}$ of a field $F$ is the smallest algebraically closed field containing $F$ . It exists and is unique up to (non-canonical) isomorphism. For example, $\overline{\mathbb{Q}} \subset \mathbb{C}$ consists of all algebraic numbers, and $[\overline{\mathbb{Q}}:\mathbb{Q}] = \infty$ .