Download Algebraische Zahlentheorie [Lecture notes] by Rainer Vogt PDF

By Rainer Vogt

Downloaded from
Sommersemester 2010

Show description

Read Online or Download Algebraische Zahlentheorie [Lecture notes] PDF

Similar algebraic geometry books

Computer Graphics and Geometric Modelling: Mathematics

Potentially the main entire evaluate of special effects as noticeable within the context of geometric modelling, this quantity paintings covers implementation and conception in an intensive and systematic type. special effects and Geometric Modelling: arithmetic, includes the mathematical heritage wanted for the geometric modeling subject matters in special effects coated within the first quantity.

Infinite Dimensional Lie Groups in Geometry and Representation Theory: Washington, DC, USA 17-21 August 2000

This booklet constitutes the complaints of the 2000 Howard convention on "Infinite Dimensional Lie teams in Geometry and illustration Theory". It provides a few very important fresh advancements during this sector. It opens with a topological characterization of standard teams, treats between different themes the integrability challenge of varied endless dimensional Lie algebras, provides colossal contributions to big topics in glossy geometry, and concludes with fascinating purposes to illustration thought.

Foundations of free noncommutative function theory

During this ebook the authors advance a idea of loose noncommutative features, in either algebraic and analytic settings. Such capabilities are outlined as mappings from sq. matrices of all sizes over a module (in specific, a vector house) to sq. matrices over one other module, which admire the dimensions, direct sums, and similarities of matrices.

Extra resources for Algebraische Zahlentheorie [Lecture notes]

Sample text

Das Minimalpolynom f von α ist separabel und hat daher nur verschiedene Nullstellen. Es folgt direkt aus der Definition, dass D(f ) = 0. 7 folgt DF/K (1, α, α2 , . . 4 auch DF/K (β1 , . . , βn ) = 0. Wir wollen nun eine Methode zur Berechnung von DF/K (1, α, . . , αn−1 ) angeben, f¨ ur die man α nicht explizit zu kennen braucht. 10 Satz: Sei F = K(α), α separabel u ¨ber K n n−1 X + an−1 X + . . + a1 X + a0 . Dann gilt  p0 p1  p1 p2  DF/K (1, α, . . , αn−1 ) = det  ..  . mit Minimalpolynom f = p2 p3 ..

N ) = det(σi (αj ))2 . Beweis: DF/K (α1 , . . , αn ) = det(SF/K (αi · αj )) = det( k σk (αi · αj )) = det( k σk (αi ) · σk (αj )) = det((σk (αi ))i,k · (σk (αj ))k,j ) = det(σk (αj ))2 . 6 Definition: Sei f ∈ K[X] ein Polynom vom Grad n und Leitkoeffizin ent c. Sei f = c · dann nennt man i=1 (X − αi ) seine Faktorisierung in Linearfaktoren in K[X], D(f ) = 1≤i

10 sind damit alle ci ganz u ¨ber R. Da ci ∈ K und R ganz abgeschlossen in K ist, sind alle ci ∈ R, also f ∈ R[X]. 18 Aufgabe: (1) Zeigen Sie: Sei R ganz abgeschlossener Integrit¨atsring und K sein Quotientenk¨orper. Sei f ∈ R[X] normiert und seien g, h ∈ K[X] normierte Polynome, so dass f =g·h Dann gilt g, h ∈ R[X]. 30 in K[X]. 17 mit Hilfe von (1). 19 Satz: Sei R ⊂ F ganze Ringerweiterung und F ein K¨orper. Dann ist auch R ein K¨orper. Beweis: Sei a = 0 aus R. Da a1 ∈ F ganz u ullt es eine Ganz¨ber R ist, erf¨ heitsgleichung mit Koeffizienten ci ∈ R 1 a n + cn−1 n−1 1 a + .

Download PDF sample

Rated 4.06 of 5 – based on 35 votes