algebraic number theory

E530317

branch of mathematics subfield of number theory

Algebraic number theory is a branch of mathematics that studies algebraic structures related to algebraic integers and number fields, focusing on properties of integers through tools from abstract algebra.

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Statements (68)

Predicate	Object
instanceOf	branch of mathematics ⓘ subfield of number theory ⓘ
aimsTo	generalize properties of integers to number fields ⓘ understand arithmetic of algebraic numbers ⓘ
centralConcept	Galois group of a number field ⓘ algebraic integer ⓘ class group ⓘ completion of fields ⓘ ideal ⓘ localization of rings ⓘ number field ⓘ prime ideal ⓘ unit group ⓘ
fieldOfStudy	Dedekind domains NERFINISHED ⓘ Diophantine equations NERFINISHED ⓘ Dirichlet L-functions NERFINISHED ⓘ Dirichlet characters NERFINISHED ⓘ Galois cohomology NERFINISHED ⓘ Galois representations NERFINISHED ⓘ Iwasawa theory NERFINISHED ⓘ L-functions NERFINISHED ⓘ Minkowski theory ⓘ algebraic integers ⓘ arithmetic geometry ⓘ class field theory ⓘ class groups ⓘ elliptic curves over number fields ⓘ global fields ⓘ ideal class groups ⓘ ideal theory ⓘ local fields ⓘ modular forms in number theory ⓘ number fields ⓘ p-adic numbers ⓘ prime decomposition in number fields ⓘ ramification theory ⓘ ring of integers of a number field ⓘ units in number fields ⓘ valuation theory ⓘ zeta functions of number fields ⓘ
historicalDevelopment	19th century ⓘ
notableContributor	André Weil NERFINISHED ⓘ David Hilbert NERFINISHED ⓘ Emil Artin NERFINISHED ⓘ Ernst Kummer NERFINISHED ⓘ Helmut Hasse NERFINISHED ⓘ John Tate NERFINISHED ⓘ Kenku Iwasawa NERFINISHED ⓘ Kurt Hensel NERFINISHED ⓘ Leopold Kronecker NERFINISHED ⓘ Richard Dedekind NERFINISHED ⓘ
relatedTo	analytic number theory ⓘ arithmetic geometry ⓘ representation theory ⓘ
studiesPropertyOf	algebraic extensions of the rationals ⓘ class numbers ⓘ factorization in rings of integers ⓘ integers ⓘ norms and traces in number fields ⓘ prime numbers ⓘ unit groups of number fields ⓘ
usesTool	Galois theory ⓘ algebraic geometry ⓘ commutative algebra ⓘ field theory ⓘ group theory ⓘ homological algebra ⓘ module theory ⓘ

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Fermat's theorem on sums of two squares → usedIn → algebraic number theory ⓘ