algebraic number theory
Algebraic Number Theory
This entry is a of to entries on algebraic number theory in Planetmath (therefore to be always under construction). It is the hope of the author(s) that someday this can be used as a “graduate text” to learn the subject by reading the individual entries listed here. Each a brief description of the concepts, which is expanded in the entries. Some of the concepts might be missing in Planetmath as of today (please consider writing an entry on them!). In to organize the entry in sections, we followed the main reference [Mar].
The entry number theory contains a nice introduction to the broad subject. From very early on, mathematicians have tried to understand the integer solutions of polynomial equations (e.g. Pythagorean triples). One of the main motivational examples for the subject is Fermat’s Last Theorem (when does have integer solutions?). The study of integer solutions immediately leads to the study of algebraic numbers (see (http://planetmath.org/Y2X32) for an example). Algebraic number theory is the study of algebraic numbers, their properties and their applications.
2 Number Fields and Rings of Integers
Norm and trace of an algebraic number. See also this entry (http://planetmath.org/NormAndTraceOfAlgebraicNumber). One also can take the norm of an ideal.
The ring of integers of a number field is finitely generated over (http://planetmath.org/RingOfIntegersOfANumberFieldIsFinitelyGeneratedOverMathbbZ).
Euclidean number fields.
3 Decomposition of Prime Ideals
It is a well-known fact that the ring of integers of a number field is a Dedekind domain.
Notice that the ring of integers of a number field is not necessarily a PID nor a UFD (see example of ring which is not a UFD). However, every fractional ideal in a Dedekind domain factors uniquely as a product of powers of prime ideals. In particular, the ideals of factor uniquely as a product of prime ideals.
Let be an extension of number fields. Let be a prime ideal of , then is an ideal of . What is the factorization of into prime ideals of ? Read about splitting and ramification in number fields and Galois extensions for a detailed explanation and definitions of the terminology.
See the entry ramification of archimedean places for the case of infinite places.
An important example: prime ideal decomposition in quadratic extensions of (http://planetmath.org/PrimeIdealDecompositionInQuadraticExtensionsOfMathbbQ).
Another important case: prime ideal decomposition in cyclotomic extensions of (http://planetmath.org/PrimeIdealDecompositionInCyclotomicExtensionsOfMathbbQ).
More generally, read calculating the splitting of primes.
4 Ideal Class Groups
The ideal class group of a number field is the quotient group of all fractional ideals modulo principal fractional ideals. In some sense, it measures the arithmetic complexity of the number field (how far is from being a PID). The class number of , denoted by , is the of . See topics on ideal class groups and discriminants for a detailed exposition.
5 The Unit Group
The unit group of a number field is the group of units of the ring of integers , and it is usually denoted by .
An application of Dirichlet’s unit theorem: units of quadratic fields.
6 Zeta Functions and -functions
7 Class Field Theory
8 Local Fields
Definition of local field (http://planetmath.org/LocalField).
Let be a valuation of the field (see the entry valuation (http://planetmath.org/Valuation) for a comprehensive introduction). The completion (http://planetmath.org/Completion) of with respect to is a local field. For example, is the completion of with respect to the -adic valuation (http://planetmath.org/PAdicValuation).
Read also about discrete valuation rings.
Hensel’s lemma provides a criterion to prove the existence of roots of polynomials in local fields. See also examples for Hensel’s lemma.
9 Galois Representations
10 Elliptic Curves
Elliptic curves are, essentially, equations of the form . Read the entry on the arithmetic of elliptic curves for a full account of this beautiful theory.
11 Modular Forms
- Mar Daniel A. Marcus, Number Fields, Springer, New York.
Note: If you would like to contribute to this entry, please send an email to the author (alozano).
|Title||algebraic number theory|
|Date of creation||2013-03-22 15:08:05|
|Last modified on||2013-03-22 15:08:05|
|Last modified by||alozano (2414)|