Quadratic field

Real quadratic field

A real quadratic field $𝐾 = ℚ (\sqrt 𝑑)$ is a quadratic field where $𝑑 > 0$ , #m/def/num/alg and hence signature $(𝑟 1, 𝑟 2) = (2, 0)$ .

Properties

The group of units is ${\pm 𝑢 𝑚 : 𝑚 \in ℤ}$ for the fundamental unit $𝑢 \in O \times 𝐾$ , uniquely determined by $𝑢 > 1$ .¹ See Fundamental unit of a real quadratic field.

Proof

^P1 easily follows from Dirichlet's unit theorem, since we have

O 𝐾 = {1, - 1} \times ℤ

as required.

Examples

$ℚ (\sqrt 223)$ .

#state/tidy | #lang/en | #SemBr

To find a fundamental unit, show that if $𝑣 > 1$ for $𝑣 \in O \times 𝐾$ , then $𝑣 \geq 𝑢$ . ↩