Resolvent Cubic
In algebra, a resolvent cubic is one of several distinct, although related, cubic polynomials defined from a monic polynomial of degree four: :P(x)=x^4+a_3x^3+a_2x^2+a_1x+a_0. In each case: * The coefficients of the resolvent cubic can be obtained from the coefficients of using only sums, subtractions and multiplications. * Knowing the roots of the resolvent cubic of is useful for finding the roots of itself. Hence the name “resolvent cubic”. * The polynomial has a multiple root if and only if its resolvent cubic has a multiple root. Definitions Suppose that the coefficients of belong to a field whose characteristic is different from . In other words, we are working in a field in which . Whenever roots of are mentioned, they belong to some extension of such that factors into linear factors in . If is the field of rational numbers, then can be the field of complex numbers or the field of algebraic numbers. In some cases, the concept of resolvent cubic i ...
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