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Difference between revisions of "Lagrange theorem"

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For Lagrange's theorem in differential calculus see [[Finite-increments formula|Finite-increments formula]].
 
For Lagrange's theorem in differential calculus see [[Finite-increments formula|Finite-increments formula]].
  
Lagrange's theorem in group theory: The order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057250/l0572501.png" /> of any [[Finite group|finite group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057250/l0572502.png" /> is divisible by the order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057250/l0572503.png" /> of any subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057250/l0572504.png" /> of it. The theorem was actually proved by J.L. Lagrange in 1771 in the study of properties of permutations in connection with research on the solvability of algebraic equations in radicals.
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Lagrange's theorem in group theory: The order $|G|$ of any [[Finite group|finite group]] $G$ is divisible by the order $|H|$ of any subgroup $H$ of it. The theorem was actually proved by J.L. Lagrange in 1771 in the study of properties of permutations in connection with research on the solvability of algebraic equations in radicals.
  
 
====References====
 
====References====
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Lagrange's theorem on congruences: The number of solutions of the [[Congruence|congruence]]
 
Lagrange's theorem on congruences: The number of solutions of the [[Congruence|congruence]]
 
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$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057250/l0572505.png" /></td> </tr></table>
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a_0 x^n + a_1 x^{n-1} + \cdots + a_n \equiv 0 \pmod p , \ \ \  a_0 \not\equiv 0 \pmod p
 
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$$
modulo a prime number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057250/l0572506.png" /> does not exceed its degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057250/l0572507.png" />. This was proved by J.L. Lagrange (see [[#References|[1]]]). It can be generalized to polynomials with coefficients from an arbitrary integral domain.
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modulo a prime number $p$ does not exceed its degree $n$. This was proved by J.L. Lagrange (see [[#References|[1]]]). It can be generalized to polynomials with coefficients from an arbitrary integral domain.
  
 
====References====
 
====References====

Latest revision as of 21:05, 11 October 2014

For Lagrange's theorem in differential calculus see Finite-increments formula.

Lagrange's theorem in group theory: The order $|G|$ of any finite group $G$ is divisible by the order $|H|$ of any subgroup $H$ of it. The theorem was actually proved by J.L. Lagrange in 1771 in the study of properties of permutations in connection with research on the solvability of algebraic equations in radicals.

References

[1] M.I. Kargapolov, J.I. [Yu.I. Merzlyakov] Merzljakov, "Fundamentals of the theory of groups" , Springer (1979) (Translated from Russian)


Comments

References

[a1] P. Hall, "The theory of groups" , Macmillan (1959) pp. Chapt. 10
[a2] B.L. van der Waerden, "Algebra" , 2 , Springer (1967) (Translated from German)

Lagrange's theorem on congruences: The number of solutions of the congruence $$ a_0 x^n + a_1 x^{n-1} + \cdots + a_n \equiv 0 \pmod p , \ \ \ a_0 \not\equiv 0 \pmod p $$ modulo a prime number $p$ does not exceed its degree $n$. This was proved by J.L. Lagrange (see [1]). It can be generalized to polynomials with coefficients from an arbitrary integral domain.

References

[1] J.L. Lagrange, "Nouvelle méthode pour résoudre les problèmes indéterminés en nombres entièrs" J.A. Serret (ed.) , Oeuvres , 2 , G. Olms, reprint (1973) pp. 653–726
[2] I.M. Vinogradov, "Elements of number theory" , Dover, reprint (1954) (Translated from Russian)

S.A. Stepanov

Comments

References

[a1] B.L. van der Waerden, "Algebra" , 2 , Springer (1971) (Translated from German)

Lagrange's theorem on the sum of four squares: Any natural number can be represented as the sum of four squares of integers. This was established by J.L. Lagrange [1]. For a generalization of Lagrange's theorem see Waring problem.

References

[1] J.L. Lagrange, "Démonstration d'un théorème d'arithmétique" J.A. Serret (ed.) , Oeuvres , 3 , G. Olms, reprint (1973) pp. 187–201
[2] J.-P. Serre, "A course in arithmetic" , Springer (1973) (Translated from French)

S.M. Voronin

Comments

References

[a1] G.H. Hardy, E.M. Wright, "An introduction to the theory of numbers" , Oxford Univ. Press (1979) pp. Chapt. 23

Lagrange's theorem on continued fractions: Any continued fraction that represents a quadratic irrationality is periodic. This was established by J.L. Lagrange [1].

References

[1] J.L. Lagrange, "Sur la solution des problèmes indéterminés du second degré" J.A. Serret (ed.) , Oeuvres , 2 , G. Olms, reprint (1973) pp. 376–535
[2] A.Ya. Khinchin, "Continued fractions" , Univ. Chicago Press (1964) pp. Chapt. II, §10 (Translated from Russian)

S.M. Voronin

Comments

References

[a1] G.H. Hardy, E.M. Wright, "An introduction to the theory of numbers" , Oxford Univ. Press (1979) pp. Chapt. 23
How to Cite This Entry:
Lagrange theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lagrange_theorem&oldid=33554
This article was adapted from an original article by N.N. Vil'yams (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article