Quadratic field

Real quadratic field

A real quadratic field 𝐾 =ℚ(√𝑑) is a quadratic field where 𝑑 >0, alg and hence signature (𝑟1,𝑟2) =(2,0).

Properties

1. The group of units is { ±𝑢𝑚 :𝑚 ∈ℤ} for the fundamental unit 𝑢 ∈O×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

O𝐾={1,−1}×ℤ

as required.

Examples

• Rationals adjoin sqrt(223).

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Footnotes

1. To find a fundamental unit, show that if 𝑣 >1 for 𝑣 ∈O×𝐾, then 𝑣 ≥𝑢. ↩