Ring of integers of a number field

Minkowski’s bound

Let be the ring of integers of a number field . Then any ideal contains a nonzero element such that

where Minkowski’s bound depends only on and is given by

where

• is the discriminant;

• is the degree of , and is the signature of .¹

Proof sketch

proof

We apply Minkowski’s convex body theorem to the convex symmetric compact region

which looks like

in the complex case.

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Footnotes

1. 2022. Algebraic number theory course notes, ¶¶3.9–3.12, pp. 63–64 ↩