The fundamental theorem of a field of mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

 is the theorem

Theorem

In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms...

 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.

For example,the fundamental theorem of calculus

Fundamental theorem of calculus

The first part of the theorem, sometimes called the first fundamental theorem of calculus, shows that an indefinite integration can be reversed by a differentiation...

 gives the relationship between differential calculus

Differential calculus

In mathematics, differential calculus is a subfield of calculus concerned with the study of the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus....

 and integral calculus, which are two distinct branches that were not obviously related.

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

Number theory

Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...

.

The mathematical literature sometimes refers to the fundamental lemma of a field. The term lemma

Lemma (mathematics)

In mathematics, a lemma is a proven proposition which is used as a stepping stone to a larger result rather than as a statement in-and-of itself...

 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 lemmas

• Fundamental lemma of calculus of variations
  Fundamental lemma of calculus of variations
  In mathematics, specifically in the calculus of variations, the fundamental lemma in the calculus of variations is a lemma that is typically used to transform a problem from its weak formulation into its strong formulation .-Statement:A function is said to be of class C^k if it is k-times...
  
  
• Fundamental lemma of Langlands and Shelstad
• Fundamental lemma of sieve theory
  Fundamental lemma of sieve theory
  In number theory, the fundamental lemma of sieve theory is any of several results that systematize the process of applying sieve methods to particular problems...
  
  
• 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 algebra
  The fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root...
  
  
• Fundamental theorem of arithmetic
  Fundamental theorem of arithmetic
  In number theory, the fundamental theorem of arithmetic states that any integer greater than 1 can be written as a unique product of prime numbers...
  
  
• Fundamental theorem of calculus
  Fundamental theorem of calculus
  The first part of the theorem, sometimes called the first fundamental theorem of calculus, shows that an indefinite integration can be reversed by a differentiation...
  
  
• Fundamental theorem of curves
  Fundamental theorem of curves
  In differential geometry, the fundamental theorem of curves states that any regular curve with non-zero curvature has its shape completely determined by its curvature and torsion....
  
  
• Fundamental theorem of cyclic groups
  Fundamental theorem of cyclic groups
  In abstract algebra, the fundamental theorem of cyclic groups states that every subgroup of a cyclic group is cyclic. Moreover, the order of any subgroup of a cyclic group G\, of order n\, is a divisor of n\,, and for each positive divisor k\, of n\, the group G\, has exactly one subgroup of order...
  
  
• Fundamental theorem of surfaces
• Fundamental theorem of finitely generated abelian groups
• Fundamental theorem of Galois theory
  Fundamental theorem of Galois theory
  In mathematics, the fundamental theorem of Galois theory is a result that describes the structure of certain types of field extensions.In its most basic form, the theorem asserts that given a field extension E /F which is finite and Galois, there is a one-to-one correspondence between its...
  
  
• Fundamental theorem on homomorphisms
  Fundamental theorem on homomorphisms
  In abstract algebra, the fundamental theorem on homomorphisms, also known as the fundamental homomorphism theorem, relates the structure of two objects between which a homomorphism is given, and of the kernel and image of the homomorphism....
  
  
• Fundamental theorem of linear algebra
  Fundamental theorem of linear algebra
  In mathematics, the fundamental theorem of linear algebra makes several statements regarding vector spaces. These may be stated concretely in terms of the rank r of an m×n matrix A and its singular value decomposition:A=U\Sigma V^T\...
  
  
• Fundamental theorem of projective geometry
• Fundamental theorem of Riemannian geometry
  Fundamental theorem of Riemannian geometry
  In Riemannian geometry, the fundamental theorem of Riemannian geometry states that on any Riemannian manifold there is a unique torsion-free metric connection, called the Levi-Civita connection of the given metric...
  
  
• Fundamental theorem of vector analysis
• Fundamental theorem of linear programming
  Fundamental theorem of linear programming
  In applied mathematics, the fundamental theorem of linear programming, in a weak formulation, states that the maxima and minima of a linear function over a convex polygonal region occur at the region's corners...
  
  
• Central limit theorem of statistics
  Central limit theorem
  In probability theory, the central limit theorem states conditions under which the mean of a sufficiently large number of independent random variables, each with finite mean and variance, will be approximately normally distributed. The central limit theorem has a number of variants. In its common...
  
   ("Central" is used here in the same sense as "fundamental")

Non-mathematical fundamental theorems

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

• Fundamental theorem of arbitrage-free pricing
  Fundamental theorem of arbitrage-free pricing
  The fundamental theorems of arbitrage/finance provide necessary and sufficient conditions for a market to be arbitrage free and for a market to be complete. An arbitrage opportunity is a way of making money with no initial investment without any possibility of loss...
  
  
• Fisher's fundamental theorem of natural selection
  Fisher's fundamental theorem of natural selection
  In population genetics, R. A. Fisher's fundamental theorem of natural selection was originally stated as:Or, in more modern terminology:- History :...
  
  
• Fundamental theorems of welfare economics
  Fundamental theorems of welfare economics
  There are two fundamental theorems of welfare economics. The first states that any competitive equilibrium or Walrasian equilibrium leads to a Pareto efficient allocation of resources. The second states the converse, that any efficient allocation can be sustainable by a competitive equilibrium...
  
  
• Fundamental equations of thermodynamics
• Fundamental theorem of poker
  Fundamental theorem of poker
  The fundamental theorem of poker is a principle first articulated by David Sklansky that he believes expresses the essential nature of poker as a game of decision-making in the face of incomplete information....