On the field of origin of an ideal

H. B. Mann
1950 Canadian Journal of Mathematics - Journal Canadien de Mathematiques  
IN this paper we shall consider integral ideals in finite algebraic extensions (S> Si» . . .) °f the field of rational numbers. Two ideals a, b in the same field S are sa^ to be equal if and only if they contain the same numbers. Let %\D S 2 and let 31 be an ideal in S2. The numbers of 21 generate an ideal a in Si and it is known that the intersection a Pi 82= 91. (See for instance Hecke, Théorie der algebraischen Zahlen, § 37). Also if a C Si an d & C 82 generate the same ideal in a field
more » ... eal in a field containing Si an d S2 then they must generate the same ideal in Si^S2 an d thus in every field containing Si and S2. We shall therefore call two ideals a and b equal if they generate the same ideal in a field containing all the numbers of a and of b. Two such ideals may therefore be denoted by the same symbol and we shall speak of an ideal a without regard to a particular field. An ideal a will be said to be contained in a field S if it may be generated by numbers in S î hi other words, if it has a basis in S-It seems natural to try to characterize those fields which contain a given ideal a, and in this paper we shall find such a characterization at least in the case that a power of a is a prime ideal in some extension of S-A necessary and sufficient condition for an ideal a to be contained in a given field S will be derived in the case that a is an ideal of order 1, as defined in this paper. For prime ideals of order greater than 1 a necessary and sufficient condition will also be given. From now on we shall consider finite algebraic extensions (Si, • • •) over a field Si itself a finite algebraic extension over the field of rational numbers. Admissible subfields of Si ar e those containing S-Throughout the paper only fields containing S will be considered. Consider an ideal a C Si-Either a is not contained in any admissible subfield of Si or Si must contain an admissible subfield S2 which has the property that a is in S2 but not in any admissible subfield of S2. We therefore define: DEFINITION 1. If a is in Si but not in any proper admissible subfield of Si then a is said to originate in Si over S-Consider Si ^82 and let a be an ideal in Si-The numbers of a which lie in S2 form an ideal SI in S2. This ideal 21 is said to correspond in S2 to the ideal a. The ideal 21 depends only on a but not on SI-DEFINITION 2. If 31 C S corresponds to a in Si and (1) 51 = a e c, (a, c) = 1 then a is said to be of order e with respect to S-
doi:10.4153/cjm-1950-002-3 fatcat:umsdapr6x5aehhnggpxitq33vu