Skip to content

Download Algebra und Zahlentheorie [Lecture notes] by Walter Gubler PDF

By Walter Gubler

Show description

Read or Download Algebra und Zahlentheorie [Lecture notes] PDF

Best cryptography books

The Code Book: The Evolution of Secrecy from Mary, Queen of Scots to Quantum Cryptography

Codes have determined the fates of empires, nations, and monarchies all through recorded heritage. Mary, Queen of Scots used to be positioned to demise by way of her cousin, Queen Elizabeth, for the excessive crime of treason after spymaster Sir Francis Walsingham cracked the key code she used to speak together with her conspirators.

Codes and Curves (Student Mathematical Library, Volume 7)

While details is transmitted, blunders are inclined to take place. This challenge has turn into more and more vital as super quantities of data are transferred electronically each day. Coding idea examines effective methods of packaging information in order that those mistakes could be detected, or maybe corrected.
The conventional instruments of coding concept have come from combinatorics and staff thought. because the paintings of Goppa within the past due Seventies, besides the fact that, coding theorists have additional ideas from algebraic geometry to their toolboxes. particularly, by way of re-interpreting the Reed-Solomon codes as coming from comparing services linked to divisors at the projective line, you possibly can see how to find new codes in response to different divisors or on different algebraic curves. for example, utilizing modular curves over finite fields, Tsfasman, Vladut, and Zink confirmed that you'll be able to outline a series of codes with asymptotically higher parameters than any formerly recognized codes.
This e-book relies on a chain of lectures the writer gave as a part of the IAS/Park urban arithmetic Institute (Utah) software on mathematics algebraic geometry. right here, the reader is brought to the intriguing box of algebraic geometric coding conception. providing the fabric within the similar conversational tone of the lectures, the writer covers linear codes, together with cyclic codes, and either bounds and asymptotic bounds at the parameters of codes. Algebraic geometry is brought, with specific awareness given to projective curves, rational capabilities and divisors. the development of algebraic geometric codes is given, and the Tsfasman-Vladut-Zink outcome pointed out above is mentioned.

Advances in Information Security and Its Application: Third International Conference, ISA 2009, Seoul, Korea, June 25-27, 2009. Proceedings (Communications in Computer and Information Science)

Welcome to the 3rd overseas convention on info safety and Ass- ance (ISA 2009). ISA 2009 was once the main complete convention interested by some of the facets of advances in details protection and coverage. the idea that of protection and insurance is rising swiftly as an exhilarating new paradigm to supply trustworthy and secure lifestyles companies.

Additional info for Algebra und Zahlentheorie [Lecture notes]

Example text

G/N heißt Faktorgruppe . 5. Es gibt eine kanonische Projektion mit π : G → G/N, x → x ¯ := π(x) := xN = [x] Da wir in G/N repr¨ asentantenweise rechnen, folgt unmittelbar, dass π ein surjektiver Gruppenhomomorphismus ist. Weiterhin ist ker(π) = N . Dies muß allerdings noch separat bewiesen werden. Beweis: g ∈ ker(π) ⇔ gN = g¯ = e¯ = eN = N . Damit ist g = g · e ∈ N . Umgekehrt folgt aus g ∈ N aber auch gN = N . 6. Sei ϕ : G1 → G2 ein Gruppenhomomorphismus. Dann ist ker(ϕ) G1 . 5 haben wir gesehen, dass ker(ϕ) eine Untergruppe von G1 ist.

Hierbei bilden die Kongruenzklassen modulo m den Ring1 Z/mZ. Wir sagen ”a ∈ Z ist kongruent zu b ∈ Z modulo mZ” genau dann, wenn gilt: a ≡ b (mod m) :⇔ m|a − b ⇔ a − b ∈ mZ ⇔ −b + a ∈ mZ Das Ziel der beiden folgenden Abschnitte ist es, dies f¨ ur beliebige H und G zu verallgemeinern. Wir wollen also ganz allgemein lernen in ”G modulo H” zu rechnen. 1. Wir lassen uns durch das Obige leiten und definieren analog: a ∼ b (mod H) :⇔ b−1 · a ∈ H f¨ ur a, b ∈ G. Alternativ h¨ atte man an dieser Stelle auch die Definition a ∼ b (mod H) :⇔ a · b−1 ∈ H ableiten k¨onnen.

2. NEBENKLASSEN 41 Den Fall einer endlichen Ordnung haben wir damit also gezeigt. Zu pr¨ ufen ist nun nur noch der Fall ord(g) = ∞. Nach unserer anf¨ang¨ lichen Uberlegung folgt aus g n = g m immer g n−m = e und damit, da ord(g) = ∞, in diesem Fall n = m. h. unendlich viele und die Behauptung ist somit gezeigt. 14. 15 (Satz von Euler). Seien ord(G) < ∞ und g ∈ G. Dann ist g ord(G) = e. 8 folgt ord(< g >)| ord(G). 13, dass ord(g) = ord(< g >). Damit gibt es ein k ∈ N mit k · ord(g) = ord(G), so dass g ord(G) = g k·ord(g) = (g ord(g) )k = ek = e gilt.

Download PDF sample

Rated 4.42 of 5 – based on 22 votes