Branches of Mathematics
Algebraic Number Theory
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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.
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 Cross-Tradition Connections
Sources
1. Algebraic Number Theory (Wikipedia)
WikipediaOpening paragraphQuote, Opening 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.
1. Algebraic Number Theory (Wikipedia)
WikipediaFailure of unique factorizationQuote, 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.
1. Algebraic Number Theory (Wikipedia)
WikipediaGaussQuote, 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.
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.
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