ON THE BASE OF A RELATIVE NUMBER-FIELD, WITH AN APPLICATION TO THE COMPOSITION OF FIELDS*

where w1 Z S2 s SC3 * * * S SCn are rational integers. If we now consider a field relative to a given subfield of this field, fron 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 relative to its subfield, then there exist r integers 81 R2 * * a ,Sr such that every integer in the larger field can be expressed in the form 9 Y1 1 + Y2 2 + ;+ Yr r where now Y1 s Y2, * * * s Yr are integers in the subfield. Brhen, however, the nulnber of classes of the subfield is greater than one, this is not the case. SOMMER t 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 01 fiJ2 R1Q, j82n, where a)1 CD2 is the base of the subfield and N1 5 132 the base of an ideal in the subfield and Q a number of the larger field, not necessarily an integer, but such that B1Q and /325L are integers in this field. In the first part of this paper I establish a similar fortn for the base of any algebraic number-field relative to any subfield, and in the last part I apply this to the study of the discriminant of the field.