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 $A$? 



Generators of the algebra of symmetric polynomials over finite fields
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 $l > n$.


