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The resolvent cubic of a monic quartic

Definition

Let

$f (x) = x^{4} + a x^{3} + b x^{2} + c x + d \in F [x]$

be monic, and let $α_{1}, α_{2}, α_{3}, α_{4}$ be its roots in a splitting field (Polynomials that split and splitting fields of a polynomial or a family of polynomials). Put

$β_{1} = α_{1} α_{2} + α_{3} α_{4}, β_{2} = α_{1} α_{3} + α_{2} α_{4}, β_{3} = α_{1} α_{4} + α_{2} α_{3} .$

The resolvent cubic of $f$ is

$R_{f} (y) : = (y - β_{1}) (y - β_{2}) (y - β_{3}) .$

Permuting the four roots permutes the set of pairings, so the coefficients are symmetric expressions in the roots and lie in $F$ by A symmetric polynomial in the roots of a monic polynomial is a polynomial in its coefficients and lies in the base ring. The explicit coefficient formula and its discriminant identity are proved in The coefficient formula and discriminant of the quartic resolvent ↗.

Depends on

Used by

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Sources

K. Conrad, Galois Groups of Cubics and Quartics, Definition 3.1 (standard reference, not scraped)
J. S. Milne, Fields and Galois Theory, v5.10, Quartic polynomials (standard reference, not scraped)