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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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11s with 11 1 :: 1 mod 01 1 . 26) == 0, 112 == 0, 112 == 111 == Thus these exists s = 3), 112 == = (1). s 0, 1, 0, = o mod <171 11 3 -= o mod 01 2 11 3 - 11 3 == 1 mod <17 3 27 Then the required ~ s = I C lli i=l ~ . 27. 23, when the relatively prime to the corresponding then the ~i' ~ Si are is relatively prime to ~ and conversely. Proof. 28. ) * If N[crt] we find m + 07. :) m + or = ~- (1). R. 18, with = hence (~) and Oii + (Jlj (1) , (i +j). Thus + or = R. 30) D. EULER PHI-FUNCTION AND MOBIUS MU-FUNCTION For rational integers, additive cyclic group of order m.

20, 0'1 1 +crt2 (\ ... n ... 0. ,11s with 11 1 :: 1 mod 01 1 . 26) == 0, 112 == 0, 112 == 111 == Thus these exists s = 3), 112 == = (1). s 0, 1, 0, = o mod <171 11 3 -= o mod 01 2 11 3 - 11 3 == 1 mod <17 3 27 Then the required ~ s = I C lli i=l ~ . 27. 23, when the relatively prime to the corresponding then the ~i' ~ Si are is relatively prime to ~ and conversely. Proof. 28. ) * If N[crt] we find m + 07. :) m + or = ~- (1). R. 18, with = hence (~) and Oii + (Jlj (1) , (i +j). Thus + or = R. 30) D.

0ij = the Kronecker delta This constitutes an "eigenvalue" equation for n = t. We have yet to see that the Dedekind domain satisfies condition III. y S..... S ) for by using unique factorization. 9). S/yEF Define We shall show Specifically. using R-modules in F S/yER ( .. R/R·). S/y ..... S /y ) t since (S/y)m = y 2(n-l) lies in the module ( see Remar k68) . • t;ER. = (yn-l). (y n-l • Sy n-2 n-l ..... 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.

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