14 Symmetric polynomials In this section, we consider a special type of polynomials in several variables, called symmetric polynomials. This is a very classic algebraic subject, which we will encounter later in great generality in Galois theory. In 12.3, we saw that for an integral domainR, a non-zero polynomialf R[X] ∈ has no more thann = deg(f) zeros inR. ForR=Z, the fundamental theorem of algebra 26.3 tells us thatf has exactlyn = deg(f) zeros, provided that we count them with multiplicity and are willing to consider zeros in a suitable ring larger thanZ itself, for exampleC. This is also true for an arbitrary integral domain, and in§21, we will construct the necessary “extensionfields.” Even if the zeros off R[X] can only be found in a larger ringR � R, it turns ∈ ⊃ out that “symmetric expressions” in the zeros off lie inR itself. We will see how to calculate them without ever venturing outside the ground ringR. The methods we give are used by all computer algebra packages. � General polynomial of degreen We define a “general” polynomial of degreen by working in the ringR=Z[T 1,T2,. ., Tn] of polynomials in then variablesT 1,T2,...,Tn. We can also allow other rings than Z for the ring of coefficients, but for the sake of simplicity, we will not do so. The e1 e2 en (total) degree of the monomialT 1 T2 ...Tn is equal toe 1 +e 2 +e 3 +...+e n, and we define the degree deg(f) of a non-zero polynomialf R as the maximum of the ∈ degrees of the monomials inf. If all monomials inf are of the same degreed, thenf is called homogeneous of degreed. An arbitrary polynomialf R of degreed can be ∈ written asf=f 0 +f 1 +f 2 +...+f d withf k homogeneous of degreek by combining the monomials of equal degree. The general polynomialF n of degreen is the monic polynomial inR[X] with zeros the variablesT 1,T2,...,Tn: n n k n k F = (X T )(X T )...(X T ) =X + ( 1) s X − R[X]. n − 1 − 2 − n − k ∈ �k=1 The coefficientss R are called the elementary symmetric polynomials in the zeros k ∈ Ti off, and the Frenchman Fran¸coisVi`ete(1540–1603) already knew that fork= 1,2, . n, the polynomials k is equal to sk = Ti1 Ti2 ...Tik , 1 i1<i2<...i n ≤ � k≤ the sum of all products of exactlyk different zeros ofF . The polynomials R is k ∈ homogeneous of degreek, and we have s1 =T 1 +T 2 +...+T n, s2 =T 1T2 +T 1T3 +...+T 1Tn +T 2T3 +T 2T4 +...+T n 1Tn, − 49 Algebra II–§14 and sn =T 1T2T3 ...Tn . Note that, by definition, the general polynomial of degreen has coefficients in the integral domainR 0 =Z[s 1, s2, . , sn] and that itsn zeros are the variables of the extension ringR=Z[T ,T ,...,T ] R . 1 2 n ⊃ 0 � Symmetric polynomials A polynomial inR=Z[T 1,T2,...,Tn] (or, more generally, inA[T 1,T2,...,Tn] for a commutative ringA) is called symmetric (in the variablesT i) if it is invariant under all permutation of the variablesT i. Somewhat more formally, we can consider the natural action of the symmetric groupS n onR given by (σf)(T ,T ,...,T ) =f(T ,T ,...,T ) forf R,σ S . 1 2 n σ(1) σ(2) σ(n) ∈ ∈ n The symmetric polynomials inR are then the polynomials inR that are invariant under the action ofS . For everyσ S , the mapf σf is an automorphism ofR, so that n ∈ n �→ we have an inclusionS Aut(R). It follows easily that the symmetric polynomials n ⊂ form a subringR R. 0 ⊂ Exercise 1. Verify this. Since thekth elementary symmetric polynomials R is symmetric, we have the k ∈ inclusionZ[s , s , . , s ] R . The fundamental theorem for symmetric polynomials 1 2 n ⊂ 0 says that this inclusion is an equality: every symmetric polynomial is a polynomial in the elementary symmetric polynomials. This theme is elaborated further in Galois theory: polynomials with “many” symmetries are contained in “small” subrings. 14.1. Fundamental theorem. LetP R=Z[T ,T ,...,T ] be a symmetric poly- ∈ 1 2 n nomial. Then there is a unique way to writeP as a polynomial in the elementary symmetric polynomialss k. We give a constructive proof, that is, a proof that gives an algorithm to write aP as an element ofZ[s 1, s2, . , sn]. Proof. We order the monomials inP lexicographically, as in a dictionary. In other e1 e2 en words, monomialsT 1 T2 ...Tn with the highest exponente 1 are in front, if two mono- mials have the samee 1, their order is determined bye 2, and so on. Now, letc T e1 T e2 ...T en withc Z 0 be thefirst term inP , lexicographically, · 1 2 n ∈ \{ } and letd=e +e +e +...+e be its degree. Then we havee e e ... e . 1 2 3 n 1 ≥ 2 ≥ 3 ≥ ≥ n After all, if this is not the case, then through a suitable permutation of theT i, we can transform this term into one that comes earlier lexicographically and that must also be inP because of the symmetry ofP ; this gives a contradiction. Now, form the monomial e1 e2 e2 e3 e3 e4 en 1 en en Σ=s 1 − s2 − s3 − . sn −1 − sn R − ∈ 50 Algebra II–§14 of degree e1 e 2 + 2(e2 e 3) + 3(e3 e 4) +...+(n 1)(e n 1 e n) + nen − − − − − − =e +e +e +...+e =d deg(P), 1 2 3 n ≤ withfirst termT e1 T e2 ...T en (lexicographically), and considerP =P cΣ R. Be- 1 2 n 1 − ∈ cause deg(Σ) deg(P ), we have deg(P 1) deg(P ), and all monomials inP 1 come ≤ e1 e2 en≤ later, lexicographically, thanT 1 T2 ...Tn . Since onlyfinitely many different mono- mials are possible when the degree is bounded, we see that by repeatedly subtracting an element ofZ[s 1, s2, . , sn], we can reduce the polynomialP to 0. In other words, P is itself contained inZ[s 1, s2, . , sn]. To prove that there are not two ways to write a polynomial inZ[T 1,...,Tn] as a polynomial inZ[s 1, s2, . , sn], we must show that there is no non-zero polynomial g Z[X ,X ,...,X ] withg(s , s , . , s ) = 0. To do this, suppose that such a ∈ 1 2 n 1 2 n polynomial does exist. Write every monomial ing in the form e1 e2 e2 e3 e3 e4 en 1 en en cX1 − X2 − X3 − ...Xn −1 − Xn , − and consider the monomialM ing for which the correspondingn-tuple (e 1, e2, . , en) comesfirst lexicographically. When expandingg(s 1, s2, . , sn) as a polynomial in e1 e2 en Z[T1,T2,...,Tn], we see thatM leads to a term cT 1 T2 ...Tn that does not disappear, so we have a contradiction. It follows from the uniqueness of representations in terms of the elementary sym- metric polynomialss k thatZ[s 1, s2, . , sn] can be viewed as a polynomial ring in the variabless k: the elementary symmetric polynomials are algebraically independent. a1 a2 an We say that a monomials 1 s2 . sn has weighta 1 + 2a2 + 3a3 +...+ na n; more generally, the weight ofg Z[s , s , . , s ] is the maximum of the weights of ∈ 1 2 n the monomials ing. If all monomials ing have the same weightd, theng is said to be isobaric of weightd. Note that the weight ofg Z[s , s , . , s ] is nothing but ∈ 1 2 n the degree ofg as an element ofR=Z[T 1,T2,...,Tn]. The proof of 14.1 shows the following. 14.2. Corollary. A homogeneous symmetric polynomial inZ[T 1,T2,...,Tn] of de- greed can be written in a unique way as an isobaric polynomial of weightd in Z[s1, s2, . , sn]. An arbitrary symmetric polynomialP can be written as a sum of homogeneous poly- nomialsP k of degreek; the polynomialsP k are symmetric because the action ofS n on Z[T1,T2,...,Tn] leaves the degree invariant. By 14.2, the polynomialP k can be written as an isobaric polynomial of weightk in the elementary symmetric polynomialss k. � Calculations with symmetric polynomials There is a shortened notation for symmetric polynomialsP , defined as follows: for everyS n-orbit inP , we write a single representative preceded by the symbol n to � 51 Algebra II–§14 indicate that we take the sum over the monomials in theS n-orbit of the representative. In this notation, thekth elementary symmetric polynomials R is equal to k ∈ sk = T1T2T3 ...Tk. n � More generally, f withf Z[T ,T ,...,T ] is the notation for the sum of the n ∈ 1 2 n polynomials in theS -orbit off. Examples: � n 2 2 2 2 2 2 2 T1 T2 =T 1 T2 +T 1 T3 +T 2 T3 +T 1T2 +T 1T3 +T 2T3 3 � T1T2T3 =T 1T2T3 +T 1T2T4 +T 1T3T4 +T 2T3T4 4 � T1T2 =T 1T2 +T 1T3 +T 1T4 +T 2T3 +T 2T4 +T 3T4 4 � If we want to use the method from the proof of 14.1 to write a given symmetric polynomial as a polynomial in thes k, then the short notation is often useful. After all, e1 e2 en if a monomialrT 1 T2 ...Tn occurs in a symmetric polynomialf Z[T 1,T2,...,Tn], e1 e2 en ∈ then n rT1 T2 ...Tn also occurs in that polynomial. 14.3. Examples. 1. Taken 2 andP=T 2 +T 2 +...+T 2 = T 2.