Vandermonde polynomial

In algebra, the Vandermonde polynomial of an ordered set of "n" variables formula_1, named after Alexandre-Théophile Vandermonde, is the polynomial:
It is also called the Vandermonde determinant, as it is the determinant of the Vandermonde matrix.
The value depends on the order of the terms: it is an alternating polynomial, not a symmetric polynomial.
Alternating.
The defining property of the Vandermonde polynomial is that it is "alternating" in the entries, meaning that permuting the formula_5 by an odd permutation changes the sign, while permuting them by an even permutation does not change the value of the polynomial – in fact, it is the basic alternating polynomial, as will be made precise below.
It thus depends on the order, and is zero if two entries are equal – this also follows from the formula, but is also consequence of being alternating: if two variables are equal, then switching them both does not change the value and inverts the value, yielding formula_6 and thus formula_7 (assuming the characteristic is not 2, otherwise being alternating is equivalent to being symmetric).
Conversely, the Vandermonde polynomial is a factor of every alternating polynomial: as shown above, an alternating polynomial vanishes if any two variables are equal, and thus must have formula_8 as a factor for all formula_9.
Alternating polynomials.
Thus, the Vandermonde polynomial (together with the symmetric polynomials) generates the alternating polynomials.
Discriminant.
Its square is widely called the discriminant, though some sources call the Vandermonde polynomial itself the discriminant.
The discriminant (the square of the Vandermonde polynomial: formula_10) does not depend on the order of terms, as formula_11, and is thus an invariant of the "unordered" set of points.
If one adjoins the Vandermonde polynomial to the ring of symmetric polynomials in "n" variables formula_12, one obtains the quadratic extension formula_13, which is the ring of alternating polynomials.
Vandermonde polynomial of a polynomial.
Given a polynomial, the Vandermonde polynomial of its roots is defined over the splitting field; for a non-monic polynomial, with leading coefficient "a", one may define the Vandermonde polynomial as
(multiplying with a leading term) to accord with the discriminant.
Generalizations.
Over arbitrary rings, one instead uses a different polynomial to generate the alternating polynomials – see (Romagny, 2005).
Weyl character formula.
 (a vast generalization)
The Vandermonde polynomial can be considered a special case of the Weyl character formula, specifically the Weyl denominator formula (the case of the trivial representation) of the special unitary group formula_15.