View source for Lagrange theorem

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.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M.I. Kargapolov,   J.I. [Yu.I. Merzlyakov] Merzljakov,   "Fundamentals of the theory of groups" , Springer  (1979)  (Translated from Russian)</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P. Hall,   "The theory of groups" , Macmillan  (1959)  pp. Chapt. 10</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  B.L. van der Waerden,   "Algebra" , '''2''' , Springer  (1967)  (Translated from German)</TD></TR></table>

Lagrange's theorem on congruences: The number of solutions of the [[Congruence|congruence]]

<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>

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.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.M. Vinogradov,   "Elements of number theory" , Dover, reprint  (1954)  (Translated from Russian)</TD></TR></table>

''S.A. Stepanov''

====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  B.L. van der Waerden,   "Algebra" , '''2''' , Springer  (1971)  (Translated from German)</TD></TR></table>

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 [[#References|[1]]]. For a generalization of Lagrange's theorem see [[Waring problem|Waring problem]].

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

''S.M. Voronin''

====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G.H. Hardy,   E.M. Wright,   "An introduction to the theory of numbers" , Oxford Univ. Press  (1979)  pp. Chapt. 23</TD></TR></table>

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

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.Ya. Khinchin,   "Continued fractions" , Univ. Chicago Press  (1964)  pp. Chapt. II, §10  (Translated from Russian)</TD></TR></table>

''S.M. Voronin''

====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G.H. Hardy,   E.M. Wright,   "An introduction to the theory of numbers" , Oxford Univ. Press  (1979)  pp. Chapt. 23</TD></TR></table>