394 APPENDIX B. CLASSIFICATION OF REAL NUMBERS

Definition B.2.3 The elementary symmetric polynomial

sk (x1,x2, · · · ,xn) ,k = 1, · · · ,n

is the coefficient of (−1)k xn−k in the following polynomial.

(x− x1)(x− x2) · · ·(x− xn)

= xn− s1xn−1 + s2xn−2−·· ·± sn

Thuss1 = x1 + x2 + · · ·+ xn

s2 = ∑i< j

xix j, s3 = ∑i< j<k

xix jxk, . . . ,sm

= ∑i1<i2···<im

xi1xi2 · · ·xim , sn = x1x2 · · ·xn

These special elementary polynomials are symmetric because switching two of thevariables xi and x j is equivalent to switching the corresponding factors in the product(x− x1)(x− x2) · · ·(x− xn) and using the same process to collect terms which multiplyxn−k. The polynomial in x does not change.

Example B.2.4

(x− x1)(x− x2)(x− x3)

= x3− x2 (x1 + x2 + x3)+ x(x1x2 + x1x3 + x2x3)− x1x2x3.

Thus the symmetric polynomials are x1 + x2 + x3,x1x2 + x1x3 + x2x3, and x1x2x3.

Note that it follows from the above definition that

αksk (x1,x2, · · · ,xn) = sk (αx1, · · · ,αxn)

Then the following result is the fundamental theorem in the subject. It is the symmetricpolynomial theorem. This is a very remarkable theorem.

Theorem B.2.5 Let g(x1,x2, · · · ,xn) be a symmetric polynomial. Then

g(x1,x2, · · · ,xn)

equals a polynomial in the elementary symmetric polynomials.

g(x1,x2, · · · ,xn) = ∑k

aksk11 · · ·s

knn

and the ak in the commutative ring are unique.

Proof: The proof is by induction on the number of variables. If n = 1, it is obviouslytrue because s1 = x1 and g(x1) can only be a polynomial in x1. Suppose the theorem is true