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 theorems of mathematical topics

Non-mathematical fundamental theorems

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

See also


  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). Retrieved 2019-11-26.
  4. "Art of Problem Solving". 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.
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