Let the degree of be . For any two terms
of , if the first of the differences
which differs from 0, is positive, we say that is higher than . Since, of cource, the similar terms of have been merged, always one of two arbitrary terms is higher than the other. The higherness is obviously transitive. Thus there is a certain highest term
in . Then we have
In fact, if e.g. , then the term
which is obtained from by changing and with each other, would be higher than .
For proving the fundamental theorem, we form now the homogeneous polynomial
and we will show that the exponents can be determined such that the highest term in is same as in .
It is easily seen that the highest term of a product 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 is
This term coincides with the highest term of , when one determines the numbers from the equations
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 , and consequently the difference
is a homogeneous symmetric polynomial of degree having the highest term lower than in . If then
is the highest term of and one denotes
one infers as above that the difference
is a homogeneous symmetric polynomial of degree having the highest term lower than in . 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 with respect to the elementary symmetric polynomials is
Similarly, the degree of is which is ; thus one infers that the degree of is equal to . This number is also the degree of the highest term of and as well the degree of itself, with respect to .
The preceding construction implies immediately that the coefficients of are elements of the ring determined by the coefficients of . We have still to prove the uniqueness of . Let’s make the antithesis that may be represented also by another polynomial in which differs from . Forming the difference of it and we get an equation of the form
where the coefficients are distinct from zero. The equation becomes identical if one expresses in it via the indeterminates . The general term of the right hand side of the equation is a homogeneous symmetric polynomial in those indeterminates; if its highest term is 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 s 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).