A.2. THE SYMMETRIC POLYNOMIAL THEOREM 361

Axiom A.2.1 Here are the axioms for a commutative ring.

1. x+ y = y+ x, (commutative law for addition)

2. There exists 0 such that x+0 = x for all x, (additive identity).

3. For each x ∈ F, there exists −x ∈ F such that x+(−x) = 0, (existence of additiveinverse).

4. (x+ y)+ z = x+(y+ z) ,(associative law for addition).

5. xy = yx,(commutative law for multiplication). You could write this as x× y = y× x.

6. (xy)z = x(yz) ,(associative law for multiplication).

7. There exists 1 such that 1x = x for all x,(multiplicative identity).

8. x(y+ z) = xy+ xz.(distributive law).

The example of most interest here is where the commutative ring is the integers Z orQ [a1, ...,ar]. Next is a definition of what is meant by a polynomial.

Definition A.2.2 Let k ≡ (k1,k2, · · · ,kn) where each ki is a nonnegative integer.Let |k| ≡∑i ki. Polynomials of degree p in the variables x1,x2, · · · ,xn are expressions of theform

g(x1,x2, · · · ,xn) = ∑|k|≤p

akxk11 · · ·x

knn

where each ak is in a commutative ring. If all ak = 0, the polynomial has no degree. Sucha polynomial is said to be symmetric if whenever σ is a permutation of {1,2, · · · ,n},

g(xσ(1),xσ(2), · · · ,xσ(n)

)= g(x1,x2, · · · ,xn)

An example of a symmetric polynomial is s1 (x1,x2, · · · ,xn) ≡ ∑ni=1 xi. Another one is

sn (x1,x2, · · · ,xn)≡ x1x2 · · ·xn.

Definition A.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.