Last week I posted about the Fundamental Theorem of Symmetric Function Theory. Zarathustra Brady pointed me to the following alternate proof in Serge Lang’s book *Algebra*. While not as direct or useful in terms of changing basis from the $e_\lambda$`s to the $m_\lambda$`s, it is a nice, clean inductive proof that I thought was worth sharing:

Assume for contradiction that the $e_\lambda$`s do not form a basis of the space of symmetric functions. We have shown that they span the space, so there is a dependence relation: some nontrivial linear combination of $e_\lambda$`s, all necessarily of the same degree, is equal to zero. Among all such linear combinations, choose one (say $P$) that holds for the smallest possible number of variables $x_1,\ldots,x_n$. Furthermore, among the possible linear combinations for $n$ variables, choose $P$ to have minimal degree.

If the number of variables is $1$, then the only elementary symmetric functions are $x_1^k$ for some $k$, and so there is clearly no linear dependence relation. So, $n\ge 2$. Furthermore, if $P$ has degree $1$ as a polynomial in the $x_i$`s, then it can involve only $e_1$, and so it cannot be identically zero. So $P$ has degree at least $2$, in at least $2$ variables.

Now, if $P$ is divisible by $x_n$, then by symmetry it is divisible by each of the variables. So, it is divisible by $e_n$, and so we can divide the equation by $e_n$ and get a relation of smaller degree, contradicting our choice of $P$. Otherwise, if $P$ is not divisible by $x_n$, set $x_n=0$. Then we get another nontrivial relation among the $e_\lambda$ in the smaller number of variables $x_1,\ldots,x_n$, again contradicting the choice of $P$. QED!