Let $F$ be a finite field and $A$ be the algebra of symmetric polynomials on $x_1,\dots,x_n$ over $F$. What is known about the generators of $A$ except the elementary symmetric polynomials? In particular, given some power sum polynomials, how to determine whether they generate $A$?
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$\begingroup$ Two much deeper (and open) problems, even over $\mathbb{C}$, are to determine which sets of power sums form a regular sequence or generate a prime ideal. See arxiv.org/pdf/1309.1098.pdf. $\endgroup$– Richard StanleyCommented Dec 27, 2016 at 13:21
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The algebra $A$ can only be generated by power sums if the characteristic of $F$ is greater than $n$. In this case, the power sums $p_i = \sum_{j=1}^n x_j^i$ with $1 \le i \le n$ generate $A$, and any other generating system consisting of power sums needs to include $p_1,\ldots,p_n$.
These claims can be justified as follows: Since the algebra of invariants is graded, any set of generators has to include generators of the degrees given by a minimal set of generators. In our case, these degrees are $1,\ldots,n$. So a generating set consisting of power sums has to include $p_1,\ldots,p_n$. Now if the characteristic $l$ is $\le n$, then $p_l = p_1^l$, so $p_l$ isn't a new generator and, in fact, no set of power sums can contain a new generator of degree $l$. On the other hand, if $l > n$, then the Newton formulae say that the elementary symmetric polynomials can be expressed in terms of $p_1,\ldots,p_n$, so the $p_i$ are generators. Alternatively, you could argue that the Jacobian determinant of the $p_i$ is nonzero if $l > n$.
