By Azriel Rosenfeld

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Mathematik für Wirtschaftswissenschaftler 2: Lineare Algebra, Funktionen mehrerer Variablen

Mathematik gehört zu den Grundfächern für jeden Studierenden der Wirtschafts- und Sozialwissenschaften. Er benötigt Kenntnisse der research, der Linearen Algebra sowie der Funktionen einer und mehrerer Variablen. Das zweibändige Taschenbuch, hervorgegangen aus Vorlesungen des Autors an der Universität Regensburg, stellt den Studienstoff sehr anschaulich dar, unterstützt durch eine Vielzahl von Beispielen und Abbildungen.

Extra info for An Introduction to Algebraic Structures

Example text

C / ! CN / D 7! DW P whose image consists of the divisors nP P such that nP is constant on each -orbit. D/ is linear). It follows that C and CN have the same genus, and that the Riemann-Roch theorem for CN implies it for C . k/. The Riemann-Roch theorem over k al shows that the map P 7! ŒP  N ! k/ is a bijection. k/, this bijection commutes with the action of N ! k/ ! 10, Chap. IV). Regular maps of curves Let k be a perfect field. A FFINE PLANE CURVES A regular map 'W Cg1 ! K/ ! f1 ; f2 /. 4. f1 ; f2 / be a pair of regular functions on Cg1 .

C / is the group of principal divisors. C / is defined also for affine curves. C / is its ideal class group. Now consider a nonsingular projective curve of genus 1. D/ D deg D if deg D 1: 6 A closed subvariety of Pn is the zero set of a finite set of homogeneous polynomials in n C 1 variables. Such a subvariety is irreducible if it can’t be written as a union of two proper closed subvarieties. k/ is infinite but each proper closed subvariety of C is finite (cf. Fulton 1969, Chap. 6). 35 4. k/. The map P 7!

We define a prime divisor on C to be such a discrete valuation ring. C / . The group of divisors on C is the free abelian group generated by the prime divisors on C . The degree of a prime divisor dimension of the residue field of Op as a k-vector P p is the P space, and deg. p/. h/ D p which has degree zero. 12 Consider an elliptic curve E W Y 2 Z D X 3 C aXZ 2 C bZ 3 ; a; b 2 k; ¤ 0; over k. Write E2 for the affine curve Y 2 D X 3 C aX C b and kŒx; y for the ring of regular functions on E2 .