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List of theorems called fundamental

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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] Some of these are classification theorems of objects which are mainly dealt with in the field. For instance, the fundamental theorem of curves describes classification of regular curves in space up to translation and rotation.

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:

Fundamental lemmata[edit]

See also[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, ISBN 978-0-471-00005-1
  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. ISBN 978-0-19-921986-5. MR 2445243. Zbl 1159.11001.
  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. S2CID 12076544.

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