Algebraic number theory MOC

Minkowski embedding

Let be a number field with signature with real embeddings and representative unreal embeddings . The Minkowski embedding alg

is determined by where we identify .

Fundamental property

Let be the ring of integers. Then is a Classical lattice of rank , moreover it has covolume alg

where is the discriminant.¹

Proof

Suppose is an Integral basis for . It suffices to show that form a basis for . To this end, let

be the matrix containing all these embeddings of the . We now apply the following elementary row operations:

1. Add to giving

2. Multiply by giving

3. Add to giving

We see now that as defined in Discriminant of a separable extension, and thus

as required.

Norm

This generalizes by ^P1 for an ideal so that

whence we define the norm on by

so that

Properties

• Minkowski’s bound

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tidy | en | sembr

Footnotes

1. 2022. Algebraic number theory course notes, ¶3.1, p. 58 ↩