Power sum symmetric polynomial

In mathematics, specifically in commutative algebra, the power sum symmetric polynomials are a type of basic building block for symmetric polynomials, in the sense that every symmetric polynomial with rational coefficients can be expressed as a sum and difference of products of power sum symmetric polynomials with rational coefficients. However, not every symmetric polynomial with integral coefficients is generated by integral combinations of products of power-sum polynomials: they are a generating set over the "rationals," but not over the "integers."
Definition.
The power sum symmetric polynomial of degree "k" in formula_1 variables "x"1, ..., "x""n", written "p""k" for "k" = 0, 1, 2, ..., is the sum of all "k"th powers of the variables. Formally,
The first few of these polynomials are 
Thus, for each nonnegative integer formula_7, there exists exactly one power sum symmetric polynomial of degree formula_7 in formula_1 variables. 
The polynomial ring formed by taking all integral linear combinations of products of the power sum symmetric polynomials is a commutative ring.
Examples.
The following lists the formula_1 power sum symmetric polynomials of positive degrees up to "n" for the first three positive values of formula_11 In every case, formula_12 is one of the polynomials. The list goes up to degree "n" because the power sum symmetric polynomials of degrees 1 to "n" are basic in the sense of the Main Theorem stated below.
For "n" = 1:
For "n" = 2: 
For "n" = 3:
Properties.
The set of power sum symmetric polynomials of degrees 1, 2, ..., "n" in "n" variables generates the ring of symmetric polynomials in "n" variables. More specifically: 
However, this is not true if the coefficients must be integers. For example, for "n" = 2, the symmetric polynomial 
has the expression 
which involves fractions. According to the theorem this is the only way to represent formula_22 in terms of "p"1 and "p"2. Therefore, "P" does not belong to the integral polynomial ring formula_23
For another example, the elementary symmetric polynomials "e""k", expressed as polynomials in the power sum polynomials, do not all have integral coefficients. For instance, 
The theorem is also untrue if the field has characteristic different from 0. For example, if the field "F" has characteristic 2, then formula_25, so "p"1 and "p"2 cannot generate "e"2 = "x"1"x"2.
"Sketch of a partial proof of the theorem": By Newton's identities the power sums are functions of the elementary symmetric polynomials; this is implied by the following recurrence relation, though the explicit function that gives the power sums in terms of the "e""j" is complicated:
Rewriting the same recurrence, one has the elementary symmetric polynomials in terms of the power sums (also implicitly, the explicit formula being complicated):
This implies that the elementary polynomials are rational, though not integral, linear combinations of the power sum polynomials of degrees 1, ..., "n". 
Since the elementary symmetric polynomials are an algebraic basis for all symmetric polynomials with coefficients in a field, it follows that every symmetric polynomial in "n" variables is a polynomial function formula_28 of the power sum symmetric polynomials "p"1, ..., "p""n". That is, the ring of symmetric polynomials is contained in the ring generated by the power sums, formula_19 Because every power sum polynomial is symmetric, the two rings are equal.
For another system of symmetric polynomials with similar properties see complete homogeneous symmetric polynomials.