A.2. THE SYMMETRIC POLYNOMIAL THEOREM 363

Now if g1 is not constant, do for g1 what was just done for g. Obtain

g(x1,x2, · · · ,xn) = sn

(sng2 (x1,x2, · · · ,xn)+Q2 (s1, · · · ,sn−1)

)+Q(s1, · · · ,sn−1)

Continue this way, obtaining a sequence of gk till the process stops with some gm beinga constant. This must happen because the degree of gk becomes strictly smaller witheach iteration. This yields a polynomial in the elementary symmetric polynomials for{x1,x2, · · · ,xn}.

Example A.2.6 Let g(x,y) = x3 + y3. It is clear that g(x,y) = g(y,x) so g is a symmetricpolynomial. Write as a polynomial in the elementary functions.

The above proof tells how to do this. First note that x3 = s̃31 where s1 is the symmetric

polynomial associated with the single variable x. Thus p(x,y)= x3+y3−s31 where this s1 is

x+y. Then p(x,y) = x3 +y3− (x+ y)3 = −3x2y−3xy2 and this equals (−xy)(3x+3y) =−3s2s1. Thus −3s1s2 = x3 + y3− s3

1 and so g(x,y) = s31−3s1s2.

You can see that if you have a symmetric polynomial in more variables, you could usea process of reducing one variable at a time in g(x1, ...,xn−1,0) to eventually obtain thisfunction as a polynomial in the symmetric polynomials in variables {x1, ...,xn−1}.

Note that if you have ∏mj=1 (x− x j) then by definition, it is the sum of terms like

g(x1, · · · ,xm)xm−k. If you replace x with xi and sum over all i, you would obtain an ex-pression of the form ∑

mi=1 g(x1, · · · ,xm)xm−k

i which would also be a symmetric polynomial.It is of the form

g(x1, · · · ,xm)xm−k1 +g(x1, · · · ,xm)xm−k

2 + · · ·+g(x1, · · · ,xm)xm−km

so when you switch some variables in this, you get the same thing.Here is a very interesting result which I saw claimed in a paper by Steinberg and Red-

heffer on Lindermannn’s theorem which follows from the above theorem. It is a very usefulproperty of symmetric polynomials and is the main tool for proving the Lindermann Weier-strass theorem.

Theorem A.2.7 Let α1, · · · ,αn be roots of the polynomial equation

p(x)≡ anxn +an−1xn−1 + · · ·+a1x+a0 = 0 (∗)

where each ai is an integer. Then any symmetric polynomial in the quantities

anα1, · · · ,anαn

having integer coefficients is also an integer. Also any symmetric polynomial with rationalcoefficients in the quantities α1, · · · ,αn is a rational number.

Proof: Let f (x1, · · · ,xn) be the symmetric polynomial having integer coefficients.From Theorem A.2.5 it follows there are integers ak1···kn such that

f (x1, · · · ,xn) = ∑k1+···+kn≤m

ak1···kn pk11 · · · p

knn (1.3)