This chapter deals with absolute number fields and orders there of.
Definitions and vocabulary
We begin by collecting the necessary definitions and vocabulary. This is in particular important for everything related to embeddings of number fields into archimdean fields, since they are at least two (slighlty) different normalizations.
By an absolute number field we mean finite extensions of , which is of type
AnticNumberField and whose elements are of type
nf_elem. Such an absolute number field is always given in the form , where is an irreducible polynomial. We call , where is the degree the power basis of . If is any element of , then the representation matrix of is the matrix representing with respect to the power basis, that is,
Let be the signature of , that is, has real embeddings , , and complex embeddings , . In Hecke the complex embeddings are always ordered such that for . The -linear function is called the Minkowski map (or Minkowski embedding).
If is an absolute number field, then an order of is a subring of the ring of integers , which is free of rank as a -module. The natural order is called the equation order of . In Hecke orders of absolute number fields are constructed (implicitely) by specifying a -basis, which is refered to as the basis of . If is the basis of , then the matrix with
is called the basis matrix of . We call the generalized index of . In case , the determinant is in fact equal to and is called the index of . The matrix is called the Minkowski matrix of .