# List of theorems called fundamental

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]}

Likewise, the mathematical literature sometimes refers to the **fundamental lemma** of a field. The term lemma is conventionally used to denote a proven proposition which is used as a stepping stone to a larger result, rather than as a useful statement in-and-of itself.

## Fundamental theorems of mathematical topics[edit]

- 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 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"[edit]

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 lemmata[edit]

- Fundamental lemma of calculus of variations
- Fundamental lemma of Langlands and Shelstad
- Fundamental lemma of sieve theory

## See also[edit]

## References[edit]

**^**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.

## External links[edit]

- 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.