  Let S=K[x_1,...,x_n] be a polynomial ring. Denote by $p_a$ the power sum
symmetric polynomial x_1^a+...+x_n^a. We consider the following two questions:
Describe the subsets $A \subset \mathbb{N}$ such that the set of polynomials
$p_a$ with $a \in A$ generate a prime ideal in S or the set of polynomials
$p_a$ with $a \in A$ is a regular sequence in S. We produce a large families of
prime ideals by exploiting Serre's criterion for normality [4, Theorem 18.15]
with the help of arithmetic considerations, vanishing sums of roots of unity
[9]. We also deduce several other results concerning regular sequences of
symmetric polynomials.
