# Fundamental theorem

In mathematics, the **fundamental theorem** of a field is the theorem which is considered to be the most central and the important one to that field.[1] As such, its naming is not necessarily based on the difficulty of its proofs, or how often it is used.[2]

For example, the fundamental theorem of calculus gives the relationship between differential calculus and integral calculus, which are two distinct branches of calculus that were not previously obviously related. On the other hand, being "fundamental" does not necessarily mean that it is the most basic result. For example, the proof of the fundamental theorem of arithmetic requires Euclid's lemma,[3] which in turn requires Bézout's identity.[4]

The names are mostly traditional, so that for example the *fundamental theorem of arithmetic* is basic to what would now be called number theory.

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. The fundamental lemma of a field is often the same as the fundamental theorem of that field (such as the case with the fundamental lemma of the Langlands program[5]), though the two need not to be always identical,

## Fundamental lemmata

- Fundamental lemma of calculus of variations
- Fundamental lemma of Langlands and Shelstad
- Fundamental lemma of sieve theory
- Feinstein's fundamental lemma (information theory)
- Fundamental lemma of interpolation theory (numerical analysis)

## Fundamental theorems of mathematical topics

- Fundamental theorem of algebra
- Fundamental theorem of algebraic K-theory
- Fundamental theorem of algebraic number theory
- Fundamental theorem of arithmetic
- Fundamental theorem of calculus
- Fundamental theorem of curves
- Fundamental theorem of cyclic groups
- Fundamental theorem of finitely generated abelian groups
- 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 interval arithmetic
- 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 Riemannian geometry
- Fundamental theorem of surfaces
- Fundamental theorem of tessarine algebra
- Fundamental theorem of symmetric polynomials
- Fundamental theorem of topos theory
- Fundamental theorem of ultraproducts
- Fundamental theorem of vector analysis

## Non-mathematical fundamental theorems

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

## See also

## Notes

- "The Definitive Glossary of Higher Mathematical Jargon — Theorem".
*Math Vault*. 2019-08-01. Retrieved 2019-11-26. - K. D. Joshi (2001).
*Calculus for Scientists and Engineers*. CRC Press. pp. 367–8. ISBN 978-0-8493-1319-6. Retrieved 2009-03-01. - Ikenaga, Bruce (2008). "The Fundamental Theorem of Arithmetic" (PDF).
*www.math.uh.edu*. Retrieved 2019-11-26. - "Art of Problem Solving".
*artofproblemsolving.com*. Retrieved 2019-11-26. - Harrell, Eben (2009-12-08). "The Top 10 Everything of 2009 - TIME".
*Time*. ISSN 0040-781X. Retrieved 2019-11-26.

## 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.