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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[edit]

Carl Friedrich Gauss referred to the law of quadratic reciprocity as the 'fundamental theorem' of quadratic residues.^[3]

Applied or informally stated 'fundamental theorems'[edit]

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

• Holland's schema theorem, or the 'fundamental theorem of genetic algorithms'

Fundamental lemmata[edit]

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See also[edit]

References[edit]

1. ^Apostol, Tom M. (1967), Calculus, Vol. 1: One-Variable Calculus with an Introduction to Linear Algebra (2nd ed.), New York: John Wiley & Sons, ISBN978-0-471-00005-1
2. ^Hardy, G. H.; Wright, E. M. (2008) [1938]. An Introduction to the Theory of Numbers. Revised by D. R. Heath-Brown and J. H. Silverman. Foreword by Andrew Wiles. (6th ed.). Oxford: Oxford University Press. ISBN978-0-19-921986-5. MR2445243. Zbl1159.11001.
3. ^Weintraub, Steven H. (2011). 'On Legendre's Work on the Law of Quadratic Reciprocity'. The American Mathematical Monthly. 118 (3): 210. doi:10.4169/amer.math.monthly.118.03.210.

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External links[edit]

• 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.