A Classical Invitation to Algebraic Numbers and Class Fields by Harvey Cohn

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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V j , ... ]. {llill R). Let 7/t, and n be Let them be represented by a or 71 ). P. 8) [ ••• ,11. 9) as the modules generated by the elements shown. , a€lt} (Historically, the presence of one addition but two "multiplications", tlt 7L 7l't nn, and led Dedekind to define a "dual" ring [I87l] which evolved to a "lattice"). We note that if 7n. and1i. 8) 25 )J. \! , etc. 12) "L'l Ie" c ". ). Thus "greatest common divisor (gcd)" is "minimal containing set"; and "least common multiple (lcm)" is "maximal common subset".

C. (integral), furthermore -1 - at '1t * can be taken The same must be true for The order of the class group is the class number. It shall be seen (Chapter 13) that the class number is finite for an algebraic number field. It can become infinite, however, for algebraic function fields as we shall illustrate for a famous classical case, elliptic functions over ~, (see Chapter 8). For ~[l=6l (see Chapter 3) the class number is 2. 30. (2,1=6) The ideal classes are (as verified in Chapter 14 below).

24a) Thus q-l, Thus o. We can now define h. at present). 22) of ~. = ~j are linearly independent (not necessarily ~i n P :L F q Ph(x) (and likewise Proof. tl(q-l). Let Pt(x) Then if II (x t _l)ll(h/t) o. tlh is cyclic. 24a,b). Pt(x). The Mobius function Note the degree of Ph(x) is Ltll(h/t) - ¢(h). 25. is called the Frobenius autos morphism and the group it generates is the Frobenius automorphism group, a-+ a P • We say that s a P = a. 38 a belongs to p s if s q is the minimal exponent (l2. s 2.

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