Real quadratic field

Fundamental unit of a real quadratic field

Let be a Real quadratic field and let be a reduced element¹ of discriminant with simple continued fraction

with period . Then the fundamental unit of is alg

which is to say by Dirichlet’s unit theorem

Proof

proof

Monogenic case

If then one simple has

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Footnotes

1. i.e. such that the simple continued fraction is purely periodic, or equivalently, . ↩