Minkowski and Bach Bounds

The class group is generated by ideals of small norm, but how small is the whole question. The unconditional bound is far too large to use; the bound under the Riemann hypothesis is small enough to be practical, which is why almost all class group results are conditional.

The Minkowski bound

Every ideal class contains an integral ideal whose norm is below a bound depending on the discriminant, the degree and the signature.

The Bach bound

Under the Generalised Riemann Hypothesis, the class group is generated by prime ideals of norm below a bound proportional to the square of the logarithm of the discriminant.

What conditionality means

The computed group is a quotient

If the factor base does not generate, the computed group is a quotient of the true class group — the class number could be larger.

It is never too large

Missing generators can only make the computed group smaller, never bigger. This asymmetry is useful.

Verification is possible

Comparing against the analytic class number formula confirms the result unconditionally if the analytic value is computed to sufficient precision.

Practical factor base sizing

In practice the factor base is sized by a trade-off rather than by the bound directly: a larger base makes relations easier to find but the linear algebra harder. See factor base selection.

Source. Henri Cohen, A Course in Computational Algebraic Number Theory, Springer GTM 138 — 4.9. 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.