Fundamental unit of a real quadratic field

Let $𝐾 = ℚ (\sqrt 𝑑)$ be a Real quadratic field and let $𝜗 \in 𝐾$ be a reduced element¹ of discriminant $Δ 𝐾 : ℚ (𝜗) = Δ 𝐾 : ℚ$ with simple continued fraction

𝜗 = [――――――― 𝑎 0; 𝑎 1, \dots, 𝑎 𝑘]

with period $𝑘$ . Then the fundamental unit of $O 𝐾$ is #m/thm/num/alg

𝜖 = 𝑞 𝑘 - 1 + 𝑞 𝑘 𝜗,

which is to say by Dirichlet's unit theorem

O \times 𝐾 = {\pm 𝜖 𝑚 : 𝑚 \in ℤ}

Proof

#missing/proof

Monogenic case

If $𝑑 \equiv 4 2, 3$ then one simple has

𝜖 = 𝑝 𝑘 + 𝑞 𝑘 \sqrt 𝑑

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

i.e. such that the simple continued fraction is purely periodic, or equivalently, $- 1 𝜗 𝜎 > 1$ . ↩

Fundamental unit of a real quadratic field

Monogenic case

Footnotes