Lagrange theorem

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)

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

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

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