Algebraic number theory is the branch of number theory that uses the techniques of abstract algebra to study the integers, the rational numbers and their generalizations to algebraic number fields. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/
Facts
Central QuestionWhether the fundamental theorem of arithmetic, that every integer factors uniquely into primes, continues to hold in the ring of integers of a general algebraic number field, a property that can fail once the integers are generalized this way. 1 Key DebateHow far Gauss's introduction of the Gaussian integers as a ring with its own arithmetic, laid out in the Disquisitiones Arithmeticae, could be extended to general number fields once unique factorization was found to fail there, a gap that motivated the ideal-theoretic machinery the field later built to repair it. 1 Classification
Pure or Applied Algebraic Number Theory
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Source Albert-Brauer-Hasse-Noether theorem (Wikipedia)
Source Brumer-Stark conjecture (Wikipedia)
Source Emil Artin (Wikipedia)
Additional Source Wikipedia: Fermat's Last TheoremWiles's general proof section
Source Ferrero-Washington theorem (Wikipedia)
Source Gauss Sum (Wikipedia)
Source Gaussian Integer (Wikipedia)
Source Grunwald-Wang theorem (Wikipedia)
Source Helmut Hasse (Wikipedia)
Source Landau prime ideal theorem (Wikipedia)
Source Leopoldt's Conjecture (Wikipedia)
Source Principal ideal theorem (Wikipedia)
Source Reflection theorem (Wikipedia)
Sources
1. Algebraic Number Theory (Wikipedia)
WikipediaOpening paragraph
Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations.
Failure of unique factorization
An important property of the ring of rational integers Z is that it satisfies the fundamental theorem of arithmetic, that every (positive) integer has a factorization into a product of prime numbers, and this factorization is unique up to the ordering of the factors. This may no longer be true in the ring of integers O of an algebraic number field K.
Gauss
One of the founding works of algebraic number theory, the Disquisitiones Arithmeticae is a textbook of number theory written in Latin by Carl Friedrich Gauss in 1798 when Gauss was 21 and first published in 1801 when he was 24.
View the Source Wikipedia: Fermat's Last Theorem
Wikimedia FoundationIncludes: Fermat's Last Theorem, Wiles's general proof sectionQuote, Includes: Fermat's Last Theorem, Wiles's general proof section
The proof's method of identification of a deformation ring with a Hecke algebra (now referred to as an R=T theorem) to prove modularity lifting theorems has been an influential development in algebraic number theory.
Gaussian Integer (Wikipedia)
Reflection theorem (Wikipedia)
Includes: Reflection Theorem, Lead sentenceQuote, Includes: Reflection Theorem, Lead sentence
In algebraic number theory, a reflection theorem or Spiegelungssatz (German for reflection theorem, see Spiegel and Satz) is one
Gauss Sum (Wikipedia)
Includes: Gauss Sum, Lead sentenceQuote, Includes: Gauss Sum, Lead sentence
In algebraic number theory, a Gauss sum or Gaussian sum is a particular kind of finite sum of roots of unity, typically G ( χ ) :=
Brumer-Stark conjecture (Wikipedia)
Albert-Brauer-Hasse-Noether theorem (Wikipedia)
Includes: Albert-Brauer-Hasse-Noether Theorem, Lead sentenceView the Source Leopoldt's Conjecture (Wikipedia)
Wikimedia FoundationIncludes: Leopoldt's Conjecture, Lead sentenceQuote, Includes: Leopoldt's Conjecture, Lead sentence
In algebraic number theory, Leopoldt's conjecture, introduced by Heinrich-Wolfgang Leopoldt, states that the p-adic regulator of a
Ferrero-Washington theorem (Wikipedia)
Grunwald-Wang theorem (Wikipedia)
Landau prime ideal theorem (Wikipedia)
Includes: Landau Prime Ideal Theorem, Lead sentenceQuote, Includes: Landau Prime Ideal Theorem, Lead sentence
In algebraic number theory, the prime ideal theorem is the number field generalization of the prime number theorem.
Principal ideal theorem (Wikipedia)
Includes: Principal Ideal Theorem, Lead sentenceQuote, Includes: Principal Ideal Theorem, Lead sentence
ideal theorem of class field theory, a branch of algebraic number theory, says that extending ideals gives a mapping on the class
Helmut Hasse (Wikipedia)
Includes: Helmut Hasse, Lead paragraphQuote, Includes: Helmut Hasse, Lead paragraph
algebraic number theory
Emil Artin (Wikipedia)
Includes: Emil Artin, Lead paragraph [in-branch]Quote, Includes: Emil Artin, Lead paragraph [in-branch]
algebraic number theory
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