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By Harvey Cohn

From the reviews/Aus den Besprechungen: "...Für den an der Geschichte der Zahlentheorie interessierten Mathematikhistoriker ist das Buch mindestens in zweierlei Hinsicht lesenswert. Zum einen enthält der textual content eine ganze Reihe von historischen Hinweisen, zum anderen legt der Autor sehr großen Wert auf eine möglichst allseitige Motivierung seiner Darlegungen und versucht dazu, insbesondere den wichtigen historischen Schritten auf dem Weg zur Klassenkörpertheorie Rechnung zu tragen. Die Anhänge von O. Taussky bilden eine wertvolle Ergänzung des Buches. ARTINs Vorlesungen von 1932, deren Übersetzung auf einem Manuskript basiert, das die Autorin 1932 selbst aus ihrer Vorlesungsnachschrift erarbeitete und von H. HASSE durchgesehen sowie mit Hinweisen versehen wurde, dürfte für Mathematiker und Mathematikhistoriker gleichermaßen von Interesse sein..." NTM- Schriftenreihe für Geschichte der Naturwissenschaften, Technik und Medizin

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O'l 2 Since at"2 (B) • there exists an ideal 61* m= then <11. 21. 18. 24. 1 aB • acrr, etc. ar* and B/y· (1) e- 1:. or *. a *. Conversely if B/ye 1:. , We write ~/:7l l/R = R, 1:. 25. ideal group. Thus (J'( forms a group. 20. 23. 18. 0 (an R-module in F). Proof. 22) B/yJl. (B) I(y) m* may be chosen relatively prime to any preassigned non- (a) =0'l

A t ). , if then (al,aZ, ••. l3 + ... still must have a finite and the chain can go no further than the last ideal Ofn required to produce these basis elements). III. 19) . This states that if y € R', S € R, and i f for some Pi € R, n n-l (Sly) + PI (Sly) + ... 9) this equation implies 0, Sly € R. If the ascending chain condition (II) is satisfied, III can be written III'. (1. , If the fractions a(S/y)m € R Clearly if m> n (Sly) (Sly) m for all for all m~ 0), € then (1, Sly, ... ,(Sly) n-l ), a € R, S/y€ R.

S/y ..... • S /y ). but y n-2 SEt. Thus If cancellation holds y n-2 n-l S= Y t; for some S/y = C and. finally. Rings satisfying condition II are called "Noetherian rings". 4. details are not required for our purposes). (Further The designation of "Axioms" is valid since integral domain can be devised satisfying anyone of the axioms and no others. C. SUFFICIENCY OF AXIOMS We shall show now that Axioms I. II. III imply that Let denote an arbitrary ideal f (0) and f (1). az R is a Dedekind ring. Then if

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