Real quadratic field

Fundamental unit of a real quadratic field

Let 𝐾 =β„š(βˆšπ‘‘) be a Real quadratic field and let πœ— ∈𝐾 be a reduced element1 of discriminant Δ𝐾:β„š(πœ—) =Δ𝐾:β„š with simple continued fraction

πœ—=[β€•β€•β€•β€•β€•β€•β€•π‘Ž0;π‘Ž1,…,π‘Žπ‘˜]

with period π‘˜. Then the fundamental unit of O𝐾 is alg

πœ–=π‘žπ‘˜βˆ’1+π‘žπ‘˜πœ—,

which is to say by Dirichlet’s unit theorem

O×𝐾={Β±πœ–π‘š:π‘šβˆˆβ„€}

Monogenic case

If 𝑑 ≑42,3 then one simple has

πœ–=π‘π‘˜+π‘žπ‘˜βˆšπ‘‘

tidy | en | SemBr

Footnotes

  1. i.e. such that the simple continued fraction is purely periodic, or equivalently, βˆ’1πœ—πœŽ >1. ↩

Graph View

Backlinks