Minkowski embedding

Let be a number field with signature with real embeddings and representative unreal embeddings . The Minkowski embedding #m/def/num/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 #m/thm/num/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:

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

as required.

This generalizes by ^P1 for an ideal so that

whence we define the norm on by

so that

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