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

• 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 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"

Fundamental lemmata

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

See also

• Main theorem of elimination theory
• List of theorems
• Toy theorem

References

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.