# Symmetric Function Theorem

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## Theorem

Let $f$ be a polynomial in $n$ variables.

Let $f$ be of degree $r$ in each of its $n$ variables.

Then $f$ is equal to a polynomial of total degree $r$ with integer coefficients in the elementary symmetric functions:

- $ds \sum x_i \sim x_i x_j, \dotsc, \prod x_j$

and the coefficients of $f$.

## Proof

## Sources

- 1994: H.E. Rose:
*A Course in Number Theory*(2nd ed.) ... (previous) ... (next): Preface to first edition: Prerequisites: $\text {(ii)}$