9.6. SYMMETRIC FUNCTIONS445Proof of Theorem9.6.6.Since themof a fixed degreedform a basis ofthe linear spaceKSdOEx1; : : : ; xnŁ, it is immediate from the previous lemmathat theof degreedalso form a basis ofKSdOEx1; : : : ; xnŁ. Therefore, thesymmetric functionsof arbitrary degree are a basis forKSOEx1; : : : ; xnŁ.Moreover, because themcan be written asintegerlinear combina-tions of the, it follows that for any ringA, the ring of symmetric poly-nomials inAOEx1; : : : ; xnŁequalsAOE1; : : : ;nŁ.nAlgorithm for expansion of symmetric polynomials in the elementarysymmetric polynomials.The matrixTwas convenient for showingthat theform a linear basis of the vector space of symmetric polynomi-als. It is neither convenient nor necessary, however, to compute the matrixand to invert it in order to expand symmetric polynomials as linear combi-nations of the. This can be done by the following algorithm instead:LetpDPaˇxˇbe a homogeneous symmetric polynomial of degreedin variablesx1; : : : ; x
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