proof of fundamental theorem of symmetric polynomials

Let  P:=P(x1,x2,,xn) be an arbitrary symmetric polynomialMathworldPlanetmath in x1,x2,,xn.  We can assume that P is homogeneousPlanetmathPlanetmathPlanetmath (, because if  P=P1+P2++Pm  where each Pi is homogeneous and if the theorem ( is true for each Pi, it is evidently true for the sum P, too.

Let the degree ( of P be d.  For any two terms


of P, if the first of the differencesPlanetmathPlanetmath


which differs from 0, is positive, we say that M is higher than N.  Since, of cource, the terms of P have been merged, always one of two arbitrary terms is higher than the other.  The higherness is obviously transitiveMathworldPlanetmathPlanetmathPlanetmathPlanetmath (  Thus there is a certain highest term


in P.  Then we have


In fact, if e.g.  α2>α1,  then the term


which is obtained from A by changing x1 and x2 with each other, would be higher than A.

For proving the fundamental theorem (, we form now the homogeneous polynomialMathworldPlanetmathPlanetmath


and we will show that the exponents ( ij can be determined such that the highest term in Qα is same as in P.

It is easily seen that the highest term of a productMathworldPlanetmath of homogeneous symmetric polynomials is equal to the product of the highest terms of the factors.  Since the highest term of


therefore the highest term of


and thus the highest term of Qα is


This term coincides with the highest term of P, when one determines the numbers ij from the equations

{i1+i2++in=α1i2++in=α2  in=αn.

Subtracting here the second equation from the first, the third equation from the second and so on, the result is


which are nonnegative integers.  Hence we get the homogeneous symmetric polynomial


having the same highest term as P, and consequently the difference


is a homogeneous symmetric polynomial of degree d having the highest term lower than in P.  If then


is the highest term of Pα and one denotes


one infers as above that the difference


is a homogeneous symmetric polynomial of degree d having the highest term lower than in Pα.  Continuing similarly, one finally (after a finite amount of steps) shall come to a difference which is equal to 0.  Accordingly one obtains


The degree of Qα with respect to the elementary symmetric polynomials is


Similarly, the degree of Qβ is β1 which is α1; thus one infers that the degree of Q is equal to α1.  This number is also the degree of the highest term of P and as well the degree of P itself, with respect to x1.

The preceding construction implies immediately that the coefficients of G are elements of the ring determined by the coefficients of P.  We have still to prove the uniqueness of Q.  Let’s make the antithesis that P may be represented also by another polynomial in p1,p2,,pn which differs from Q.  Forming the difference of it and Q we get an equation of the form


where the coefficients are distinct from zero.  The equation becomes identical if one expresses p1,p2,,pn in it via the indeterminates x1,x2,,xn.  The general term of the right hand side of the equation is a homogeneous symmetric polynomial in those indeterminates; if its highest term is gip1λ1p2λ2pnλn,  one infers as before that


Thus, distinct addends of the sum cannot have equal highest terms.  It means that the highest term of the sum appears only in one of the addends of the sum.  This is, however, impossible, because after the substitution of xis the equation would not be identical.  Consequently, the antithesis is wrong and the whole fundamental theorem of symmetric polynomials has been proved.


  • 1 K. Väisälä: Lukuteorian ja korkeamman algebran alkeet.  Tiedekirjasto No. 17.   Kustannusosakeyhtiö Otava, Helsinki (1950).
Title proof of fundamental theorem of symmetric polynomials
Canonical name ProofOfFundamentalTheoremOfSymmetricPolynomials
Date of creation 2016-02-22 13:29:31
Last modified on 2016-02-22 13:29:31
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 14
Author pahio (2872)
Entry type Proof
Classification msc 12F10
Classification msc 13B25
Synonym proof of fundamental theorem of symmetric functions