On the base of a relative number field, with an application to the composition of fields
Transactions of the American Mathematical Society
It is a well known fact that in an algebraic number-field of degree n there exist n integers a" ■ • •, an such that every integer of the field can be expressed in the form where xx, x2, x3, ■ ■ -, xv ave rational integers. If we now consider a field relative to a given subfield of this field, from the nature of the proof of the above fact it is evident that, if the subfield in question is such that the number of classes of ideals in this field is one, and r the degree of the larger field
... larger field relative to its subfield, then there exist r integers ßx, ß2, • • -, ßr such that every integer in the larger field can be expressed in the form where now yx,y2, ---, yr are integers in the subfield. When, however, the number of classes of the subfield is greater than one, this is not the case. Sommer | has shown that in this case for a field which is of the second degree relative to a subfield of the second degree the four numbers composing the base may be taken to be tox, to2, ßx£l, ß2Q, where o>" <b2 is the base of the subfield and /3" ß2 the base of an ideal in the subfield and il a number of the larger field, not necessarily an integer, but such that ßx il and ß2 O, are integers in this field. In the first part of this paper I establish a similar form for the base of any algebraic nuniber-field relative to any subfield, and in the last part I apply this to the study of the discriminant of the field.