Class Groups, Units and Regulators

The Dirichlet Unit Theorem, Computationally

The structure of the unit group, its rank from the signature, and what computing units actually requires.

Engineering / MathematicsClass Groups, Units and Regulators2 min readKV-MATH-0594

The Dirichlet unit theorem gives the structure of the unit group completely: a finite cyclic torsion part and a free part of known rank. What it does not give is any way to find the generators.

The theorem

O^x is isomorphic to mu(K) x Z^(r1 + r2 - 1)mu(K) is the finite group of roots of unity in K.

Unit rank by signature

Rational numbersr1 = 1, r2 = 0, rank 0 — units are plus and minus one

Imaginary quadraticr1 = 0, r2 = 1, rank 0 — only roots of unity

Real quadraticr1 = 2, r2 = 0, rank 1 — one fundamental unit

Totally real cubicr1 = 3, rank 2

Generalrank = r1 + r2 - 1

Roots of unity

The torsion subgroup is cyclic and easy to compute. Its order divides a bound obtained from the field degree, and candidates are tested directly.

Computing the roots of unity

Bound the orderOnly roots of unity whose degree divides the field degree can occur, bounding the order.
Test each candidateCheck whether the cyclotomic polynomial of that order has a root in the field.
Take the largestThe group is cyclic; the largest order found generates it.

Fundamental units

A system of fundamental units is a basis of the free part. The theorem guarantees one exists but offers no construction, and finding them is the hard part.

Verification

A candidate system is fundamental when the units are independent and generate the full unit group. Independence is checked via the logarithmic embedding; generation is checked by comparing the regulator against the analytic class number formula.

Source. Henri Cohen, A Course in Computational Algebraic Number Theory, Springer GTM 138 — 4.9.2. Structural reference unverified: the source file was not available during authoring; chapter and section numbers are taken from the published edition and have not been checked against a physical copy.