Algebraische Zahlentheorie by Jürgen Neukirch

By Jürgen Neukirch

Die algebraische Zahlentheorie ist eine der traditionsreichsten und gleichzeitig heute besonders aktuellen Grunddisziplinen der Mathematik. In dem vorliegenden Buch wird sie in einem ausführlichen und weitgefaßten Rahmen abgehandelt, der sowohl die Grundlagen als auch ihre Höhepunkte enthält. Die Darstellung führt den Studenten in konkreter Weise in das Gebiet ein, läßt sich dabei von modernen Erkenntnissen übergeordneter Natur leiten und ist in vielen Teilen neu. Der grundlegende erste Teil ist mit einigen neuen Aspekten versehen, wie etwa der "Minkowski-Theorie" und einer ausführlichen Theorie der Ordnungen. Über die Grundlagen hinaus enthält das Buch eine geometrische Neubegründung der Theorie der algebraischen Zahlkörper durch die Entwicklung einer "Riemann-Roch-Theorie" vom "Arakelovschen Standpunkt", die bis zu einem "Grothendieck-Riemann-Roch-Theorem" führt, ferner eine moderne Darstellung der Klasssenkörpertheorie und schließlich eine neue Theorie der Theta-Reihen und L-Reihen, die die klassischen Arbeiten von Hecke in eine faßliche shape setzt. Das Buch ist an Studenten nach dem Vorexamen gerichtet, darüber hinaus wird es sehr bald dem Forscher als weiterweisendes Handbuch unentbehrlich sein.

Show description

Read or Download Algebraische Zahlentheorie PDF

Best number theory books

Ramanujan's Notebooks

This booklet constitutes the 5th and ultimate quantity to set up the implications claimed by way of the nice Indian mathematician Srinivasa Ramanujan in his "Notebooks" first released in 1957. even supposing all of the 5 volumes includes many deep effects, maybe the typical intensity during this quantity is larger than within the first 4.

Problem-Solving and Selected Topics in Number Theory: In the Spirit of the Mathematical Olympiads

This booklet is designed to introduce probably the most vital theorems and effects from quantity conception whereas trying out the reader’s figuring out via conscientiously chosen Olympiad-caliber difficulties. those difficulties and their options give you the reader with a chance to sharpen their talents and to use the speculation.

Primality testing for beginners

How are you going to inform even if a host is fundamental? What if the quantity has hundreds and hundreds or hundreds of thousands of digits? this query could appear summary or beside the point, yet in reality, primality exams are played whenever we make a safe on-line transaction. In 2002, Agrawal, Kayal, and Saxena responded a long-standing open query during this context by way of proposing a deterministic try out (the AKS set of rules) with polynomial operating time that exams even if a host is fundamental or now not.

Extra info for Algebraische Zahlentheorie

Example text

For n = 1 this is clear from the maximality of p , and for n 1 1 it follows by induction: A = pn-' s A + p = pA = p(pn-' s A ) p" s A jp" s A = A. , a = hls with h E pi', s 4 p . Then as = b E p n , so that ZS = 0 in Alp", and hence 5 = 0 in A l p n . Surjectivity of f : let a l s E A p r a E A, s 4 p . Then by the above, there exists an a' E A such that a = a's mod pn. , a l s mod m; lies in the image of f ' . + + + 5 + + In a local ring with maximal ideal m, every element a 4 m is a unit. Indeed, since the principal ideal ( a ) is not contained in any other maximal ideal, it has to be the whole ring.

It avoids pathological situations and is satisfied in all interesting cases, in particular for the orders in an algebraic number field. 9) Proposition.

A = I mod 8 when 81h. Now xc(f) = 0, as one sees by multiplying the sum with a symbol (3 = -1, and putting 6 = {"-I gives from which we indeed find that Eb{ h ( c - l ) -1, = t + p + . . +=p Exercise 7. Let ( a ,p ) = 1 and av = r, mod p , v = 1, . . , < p. Then the r, give a permutation n of the numbers 1, . . , p - 1. Show that sgn ~r= ( a ) . P Exercise 8. Let a, = &" - &In J5 , where E = -I,+ & Fibonacci number). If p is a prime number 2 &' = -((I,, is the 11-th # 2 , 5 , then one has P a/>=(,) modp.

Download PDF sample

Rated 4.92 of 5 – based on 30 votes