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 $|G|$ 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====
<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]]
$$
a_0 x^n + a_1 x^{n-1} + \cdots + a_n \equiv 0 \pmod p , \ \ \  a_0 \not\equiv 0 \pmod p
$$
modulo a prime number $p$ does not exceed its degree $n$. This was proved by J.L. Lagrange (see [[#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>