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 equivalence relations
• 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 interval arithmetic
• Fundamental theorem of Lebesgue integral calculus
• 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:

• 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"

There is no single fundamental theorem of statistics, but rather there are three, each of which would be the fundamental theorem if the others were not equally or even more important:

Foundational statistical theorems

• The law of large numbers
• The central limit theorem (multiple forms)
• The extreme value theorem (analogue of the central limit theorem, but for extreme values rather than central values)

There are other candidates, such as Chebyshev's inequality.

See also

• List of theorems
• Toy theorem

Notes

1. "The Definitive Glossary of Higher Mathematical Jargon — Theorem". Math Vault. 2019-08-01. Retrieved 2019-11-26.
2. K. D. Joshi (2001). Calculus for Scientists and Engineers. CRC Press. pp. 367–8. ISBN 978-0-8493-1319-6. Retrieved 2009-03-01.
3. Ikenaga, Bruce (2008). "The Fundamental Theorem of Arithmetic" (PDF). www.math.uh.edu. Retrieved 2019-11-26.
4. "Art of Problem Solving". artofproblemsolving.com. Retrieved 2019-11-26.
5. 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.