Ring of integers

The ring of integers of a number field forms a lattice

Let be a number field of degree and denote its ring of integers. Then is a lattice subgroup of of rank .¹ ring

Proof

By ^P2, we can form a -basis of algebraic integers spanning . Suppose towards contradiction that is not discrete, so there are arbitrarily small such that is nonzero. Now for each embedding we have

so for some homogenous polynomial of degree , whence becomes arbitrarily small as are made arbitrarily small. But since is an algebraic integer, so is , meaning it must be an “arbitrarily small nonzero integer”, a contradiction.

It follows that any nonzero ideal is a (full rank) sublattice of , whence is finite.

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Footnotes

1. 2022. Algebraic number theory course notes, ¶1.18, p. 14 ↩