and have equal lacunary parts. (This is because every monomial which can appear in a lacunary part must lack at least one variable, and thus can be transformed by a permutation of the variables into a monomial which contains only the variables .) But the terms of which contain only the variables are precisely the terms that survive the operation of setting to 0, so their sum equals , which is a symmetric polynomial in the variables that we shall denote by . By the inductive hypothesis, this polynomial can be written as for some . Here the doubly indexed denote the elementary symmetric polynomials in variables. Consider now the polynomial Then is a symmetric polynomial in , of the same degree as , which satisfies (the first equality holds because setting to 0 in gives , for all ). In other words, the coefficient of before each monomial which contains only the variables equals the corresponding coefficient of . As we know, this shows that the lacunary part of coincides with that of the original polynomial . Therefore the difference has no lacunary part, and is therefore divisible by the product of all variables, which equals the elementary symmetric polynomial . Then writing , the quotient is a homogeneous symmetric polynomial of degree less than (in fact degree at most ) which by the inductive hypothesis can be expressed as a polynomial in the elementary symmetric functions. Combining the representations for and one finds a polynomial representation for . The uniqueness of the representation can be proved inductively in a similar way. (It is equivalent to the fact that the polynomials are algebraically independent over the ring .) The fact that the polynomial representation is unique implies that is isomorphic to . Alternative proof The following proof is also inductive, but does not involve other polynomials than those symmetric in , and also leads to a fairly direct procedure to effectively write a symmetric polynomial as a polynomial in the elementary symmetric ones. Assume the symmetric polynomial to be homogeneous of degree ; different homogeneous components can be decomposed separately. Order the monomials in the variables lexicographically, where the individual variables are ordered , in other words the dominant term of a polynomial is one with the highest occurring power of , and among those the one with the highest power of , etc. Furthermore parametrize all products of