Fundamental Theorem of Symmetric Polynomials

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Theorem

Let $K$ be a field.

Let $f$ a symmetric polynomial over $K$.


Then $f$ can be written uniquely as a polynomial in the elementary symmetric polynomials.


Proof

Let $p \in R := K \sqbrk {x_1, \dotsc, x_n}$ be a symmetric polynomial.

Let $\sigma_1, \dotsc, \sigma_n$ denote the $n$ elementary symmetric polynomials.

Let $<$ be the degree reverse lexicographic monomial order.

Let us denote by $\map {LM} f$ the leading monomial of any $f \in R$, that is to say the unique $<$-largest monomial appearing in $f$.

Then it is straightforward to verify that $\map {LM} {\sigma_k} = x_1 \cdots x_k$ for $1 \le k \le n$:

For $k < n$, any other monomial appearing in $\sigma_k$ must contain a factor $x_j^{\alpha_1}$ with $j > k$ and $\alpha_1 > 0$.

This makes it smaller than $x_1 \cdots x_k$ with respect to $<$.

The case $k = n$ is trivial because $\sigma_n$ consists of only one monomial.


It follows the proof that $p$ may be written as a polynomial in $\sigma_1, \dotsc, \sigma_k$.

Firstly, since $p$ is symmetric, it must have the property:

$\map {LM} p = x_1^{a_1} \dotsm x_n^{a_n}$

with $a_1 \ge a_2 \ge \ldots \ge a_n$.

To see this, assume that $a_i > a_j$ for some $1 \le i < j \le n$.

The monomial:

$x_1^{a_1} \dotsm x_{i - 1}^{a_{i - 1} } \cdot x_i^{a_j} x_j^{a_i} \cdot x_{j + 1}^{a_{j + 1} } \cdots x_n^{a_n}$

must also appear in $p$, and if $a_i > a_j$ then it would be larger than $\map {LM} p$, a contradiction.

Now consider:

$\sigma := c \sigma_1^{a_1 - a_2} \cdot \sigma_2^{a_2 - a_3} \cdot \dotsm \cdot \sigma_{n - 1}^{a_{n - 1} - a_n} \cdot \sigma_n^{a_n}$

where $c$ is the coefficient of $\map {LM} p$ in $p$.


Define $\bar p := p - \sigma$.

Because:

\(\displaystyle \map {LM} \sigma\) \(=\) \(\displaystyle \prod_{i \mathop = 1}^n \map {LM} {\sigma_i^{a_i - a_{i + 1} } }\)
\(\displaystyle \) \(=\) \(\displaystyle x_1^{a_1 - a_2} \cdot \paren {x_1 x_2}^{a_2 - a_3} \cdot \dotsm \cdot \paren {x_1 \ldots x_{n - 1} }^{a_{n - 1} - a_n} \cdot \paren {x_1 \ldots x_n}^{a_n}\)
\(\displaystyle \) \(=\) \(\displaystyle x_1^{a_1} \dotsm x_n^{a_n}\)
\(\displaystyle \) \(=\) \(\displaystyle \map {LM} p\)

the leading monomials in the difference $\bar p$ cancel.


Therefore:

$\map {LM} {\bar p} < \map {LM} p$

Iterate this method, and after finitely many steps one has decomposed $p$ into a polynomial $F \in K \sqbrk {y_1, \dotsc, y_n}$ with $\map F {\sigma_1, \dotsc, \sigma_n} = p$.

The termination of this algorithm follows immediately from Dickson's Lemma.

$\blacksquare$