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Differents, discriminants and Steinitz classes
Author(s) -
Schmid Peter
Publication year - 2013
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/bds092
Subject(s) - mathematics , ideal (ethics) , discriminant , class (philosophy) , ideal class group , combinatorics , algebraic number , square (algebra) , extension (predicate logic) , interpretation (philosophy) , square root , root of unity , algebraic number field , mathematical analysis , geometry , physics , philosophy , epistemology , artificial intelligence , computer science , programming language , quantum mechanics , quantum
Let L | K be an extension of algebraic number fields. By a theorem of Hecke, the ideal class of the different L | K is a square in Cl L . Whereas the class of the discriminant L | K =N L | K ( L | K ) is the square of the Steinitz class s L | K in Cl K , which vanishes if and only if the ring R L of integers in L is a free module over R K , there is no distinguished square root for the different in general. In an attempt to give some interpretation for Hecke's theorem, we examine when there is r ∈Cl L such that r 2 =[ L | K ] and N L | K ( r )= s L | K (‘Steinitz root’). In this case, every ideal of L in the class r −1 is a free R K ‐module.