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Arithmetic topology is an area of mathematics that is a combination of algebraic number theory and topology. In the 1960s topological interpretations of class field theory were given by John Tate^[1] based on Galois cohomology, and also by Michael Artin and Jean-Louis Verdier^[2] based on Étale cohomology. Then David Mumford (and independently Yuri Manin) came up with an analogy between prime ideals and knots^[3] which was further explored by Barry Mazur^[4]^[5]. In the 1990s Reznikov^[6] and Kapranov^[7] began studying these analogies, coining the term arithmetic topology for this area of study.

^ J. Tate, Duality theorems in Galois cohomology over number fields, (Proc. Intern. Cong. Stockholm, 1962, p. 288-295).
^ M. Artin and J.-L. Verdier, Seminar on étale cohomology of number fields, Woods Hole, 1964.
^ Who dreamed up the primes=knots analogy?, neverendingbooks, lieven le bruyn's blog, may 16, 2011,
^ Remarks on the Alexander Polynomial, Barry Mazur, c.1964
^ B. Mazur, Notes on ´etale cohomology of number fields, Ann. scient. ´Ec. Norm. Sup. 6 (1973), 521-552.
^ A. Reznikov, Three-manifolds class field theory (Homology of coverings for a nonvirtually b1-positive manifold), Sel. math. New ser. 3, (1997), 361–399.
^ M. Kapranov, Analogies between the Langlands correspondence and topological quantum field theory, Progress in Math., 131, Birkhäuser, (1995), 119–151.

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v t e Number theory
Fields	Algebraic number theory (class field theory, non-abelian class field theory, Iwasawa theory, Iwasawa–Tate theory, Kummer theory) Analytic number theory (analytic theory of L-functions, probabilistic number theory, sieve theory) Geometric number theory Computational number theory Transcendental number theory Diophantine geometry (Arakelov theory, Hodge–Arakelov theory) Arithmetic combinatorics (additive number theory) Arithmetic geometry (anabelian geometry, p-adic Hodge theory) Arithmetic topology Arithmetic dynamics
Key concepts	Numbers 0 Natural numbers Unity Prime numbers Composite numbers Rational numbers Irrational numbers Algebraic numbers Transcendental numbers p-adic numbers (p-adic analysis) Arithmetic Modular arithmetic Chinese remainder theorem Arithmetic functions
Advanced concepts	Quadratic forms Modular forms L-functions Diophantine equations Diophantine approximation Irrationality measure Simple continued fractions
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