List of theorems called fundamental

Last updated September 15, 2024

In mathematics, a fundamental theorem is a theorem which is considered to be central and conceptually important for some topic. For example, the fundamental theorem of calculus gives the relationship between differential calculus and integral calculus.^[1] The names are mostly traditional, so that for example the fundamental theorem of arithmetic is basic to what would now be called number theory.^[2] Some of these are classification theorems of objects which are mainly dealt with in the field. For instance, the fundamental theorem of curves describes classification of regular curves in space up to translation and rotation.

Fundamental theorems of mathematical topics

Fundamental theorem of algebra
Fundamental theorem of algebraic K-theory
Fundamental theorem of arithmetic
Fundamental theorem of Boolean algebra
Fundamental theorem of calculus
Fundamental theorem of calculus for line integrals
Fundamental theorem of curves
Fundamental theorem of cyclic groups
Fundamental theorem of dynamical systems
Fundamental theorem of equivalence relations
Fundamental theorem of exterior calculus
Fundamental theorem of finitely generated abelian groups
Fundamental theorem of finitely generated modules over a principal ideal domain
Fundamental theorem of finite distributive lattices
Fundamental theorem of Galois theory
Fundamental theorem of geometric calculus
Fundamental theorem on homomorphisms
Fundamental theorem of ideal theory in number fields
Fundamental theorem of Lebesgue integral calculus
Fundamental theorem of linear algebra
Fundamental theorem of linear programming
Fundamental theorem of noncommutative algebra
Fundamental theorem of projective geometry
Fundamental theorem of random fields
Fundamental theorem of Riemannian geometry
Fundamental theorem of tessarine algebra
Fundamental theorem of symmetric polynomials
Fundamental theorem of topos theory
Fundamental theorem of ultraproducts
Fundamental theorem of vector analysis

Carl Friedrich Gauss referred to the law of quadratic reciprocity as the "fundamental theorem" of quadratic residues.^[3]

Applied or informally stated "fundamental theorems"

There are also a number of "fundamental theorems" that are not directly related to mathematics:

Fundamental theorem of arbitrage-free pricing
Fisher's fundamental theorem of natural selection
Fundamental theorems of welfare economics
Fundamental equations of thermodynamics
Fundamental theorem of poker
Holland's schema theorem, or the "fundamental theorem of genetic algorithms"
Glivenko–Cantelli theorem, or the "fundamental theorem of statistics"
Fundamental theorem of software engineering

Fundamental lemmata

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References

↑ Apostol, Tom M. (1967), Calculus, Vol. 1: One-Variable Calculus with an Introduction to Linear Algebra (2nd ed.), New York: John Wiley & Sons, ISBN 978-0-471-00005-1
↑ Hardy, G. H.; Wright, E. M. (2008) [1938]. An Introduction to the Theory of Numbers. Revised by D. R. Heath-Brown and J. H. Silverman. Foreword by Andrew Wiles. (6th ed.). Oxford: Oxford University Press. ISBN 978-0-19-921986-5. MR 2445243. Zbl 1159.11001.
↑ Weintraub, Steven H. (2011). "On Legendre's Work on the Law of Quadratic Reciprocity". The American Mathematical Monthly . 118 (3): 210. doi:10.4169/amer.math.monthly.118.03.210. S2CID 12076544.

External links

Media related to Fundamental theorems at Wikimedia Commons
"Some Fundamental Theorems in Mathematics" (Knill, 2018) - self-described "expository hitchhikers guide", or exploration, of around 130 fundamental/influential mathematical results and their significance, across a range of mathematical fields.

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[1] Apostol, Tom M. (1967), Calculus, Vol. 1: One-Variable Calculus with an Introduction to Linear Algebra (2nd ed.), New York: John Wiley & Sons, ISBN 978-0-471-00005-1

[2] Hardy, G. H.; Wright, E. M. (2008) [1938]. An Introduction to the Theory of Numbers. Revised by D. R. Heath-Brown and J. H. Silverman. Foreword by Andrew Wiles. (6th ed.). Oxford: Oxford University Press. ISBN 978-0-19-921986-5. MR 2445243. Zbl 1159.11001.

[3] Weintraub, Steven H. (2011). "On Legendre's Work on the Law of Quadratic Reciprocity". The American Mathematical Monthly . 118 (3): 210. doi:10.4169/amer.math.monthly.118.03.210. S2CID 12076544.

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