Fundamental theorem

The fundamental theorem of a field of mathematics is the theorem considered central to that field. The naming of such a theorem is not necessarily based on how often it is used or the difficulty of its proofs.^[1]

For example, the fundamental theorem of calculus gives the relationship between differential calculus and integral calculus, which are two distinct branches that are not obviously related. 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, which in turn requires Bézout's identity.

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, but not always, the same as the fundamental theorem of that field.

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 surfaces
• Fundamental theorem of finitely generated abelian groups
• Fundamental theorem of Galois theory
• 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 projective geometry
• Fundamental theorem of Riemannian geometry
• Fundamental theorem of symmetric polynomials
• Fundamental theorem of topos theory
• Fundamental theorem of vector analysis
• Fundamental theorem of linear programming
• Fundamental theorem of ultraproducts

Non-mathematical fundamental theorems

There are also a number of "fundamental theorems" 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"

See also

• List of theorems

Notes