each $x_{ij}^2$, $1 \le i < j \le n+1$, by quantities like $x_i+x_j$, $x_i x_j$, $(x_i+x_j)^2$, $x_i^2+x_ix_j+x_j^2$; see [@impurity Theorem  5.1, p. 54]. Such substitutions obviously do not preserve irreducibility. Furthermore, our treatment is not confined to the cases when $k = {\mathbb{R}}$ or ${\mathbb{C}}$.

The paper is organized as follows. Section \[222\] contains preliminary facts that we shall need, and freely use, in the sequel. These are quite few, and come from the elementary theories of symmetric and of homogeneous polynomials. Section \[333\] contains proofs of the main results, namely Theorems \[DDD\], \[CCDD\], and \[CM\]. These proofs make use of Lemmas \[AAA\], \[BBB\], and \[CCC\].

Preliminaries {#222}
=============

In this section, we put together definitions and simple facts pertaining to symmetric and homogeneous polynomials that we shall freely use throughout the paper. For ease of reference, we also include a simple theorem about the factorization of quadratics.

Let $A$ be any commutative ring with identity, and let $B=A[x_1,\cdots,x_n]$ be the polynomial ring in the $n$ indeterminates $x_1,\cdots,x_n$. Then the symmetric group ${\mathcal{S}}_n$ acts as permutations on the indices of $x_1,\cdots,x_n$, and hence as $A$-automorphisms on $B$. A polynomial in $B$ is called [*symmetric*]{} if it is invariant under the 