Algebraic Number Theory
The algebraic number theory is a branch of number theory , which in turn is a branch of mathematics is.
The algebraic number theory goes beyond the whole or rational numbers and considers algebraic number fields , which are finite extensions of the rational numbers. Elements of number fields are zeros of polynomials with rational coefficients. These number fields contain subsets analogous to the whole numbers, the whole rings . Wholeness rings are Dedekind rings and in many ways behave like the ring of whole numbers, but some properties take a slightly different shape. For example, there is generally no longer any clear decomposition into prime numbers, but only into prime ideals .
Algebraic number theory continues to study algebraic function fields over finite fields, the theory of which is largely analogous to the theory of number fields. Algebraic number and function fields are summarized under the name " global fields ".
It often turns out to be fruitful to consider questions “locally”, ie for each prime position individually ( local-global principle ). In the case of integers, this process leads to the p -adic numbers , more generally to local fields .
Additional terms
literature
- Jürgen Neukirch : Algebraic number theory . Springer-Verlag, Berlin 1992. ISBN 3-540-54273-6 .
- Serge Lang : Algebraic Number Theory . Springer-Verlag, 2000. ISBN 0-387-94225-4 .
- Władysław Narkiewicz : Elementary and Analytic Theory of Algebraic Numbers . Springer-Verlag, 2004. ISBN 3-540-21902-1 .
- J. Neukirch , A. Schmidt , K. Wingberg : Cohomology of number fields . Springer-Verlag, Berlin-Heidelberg-New York 1999, ISBN 3-540-66671-0 .
- Ian Stewart , David Tall : Algebraic Number Theory and Fermat's Last Theorem . 3rd edition. Natick (MA, USA) 2002. AK Peters. ISBN 1-56881-119-5
Web links
Lecture on algebraic number theory (19 sessions) by Yorck Sommerhäuser, WiSe 2003/2004 at LMU Munich in Quicktime format with simultaneous display of the presentation