DefinitionDefinition: AI-adaptedProof: Not applicableaudited 2026-08-16

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The monic resultant $Res (f, g)$ from the symmetric coefficient expression of $\prod_{i} g (x_{i})$

Definition

Let $f (t) = t^{n} + a_{1} t^{n - 1} + \dots + a_{n}$ be monic and let $g (t)$ be any polynomial over the same commutative ring. The polynomial

$\prod_{i = 1}^{n} g (x_{i})$

is symmetric in the formal variables $x_{1}, \dots, x_{n}$ . By Fundamental theorem of symmetric polynomials: unique expression as a polynomial in $e_{1}, \dots, e_{n}$ it has a unique expression $Q_{g} (e_{1}, \dots, e_{n})$ with coefficients polynomial in the coefficients of $g$ . The monic resultant of $f$ and $g$ is

$Res (f, g) : = Q_{g} (- a_{1}, a_{2}, \dots, (- 1)^{n} a_{n}) .$

For $n = 0$ , the product is empty and $Res (1, g) = 1$ .

Depends on

Used by

For monic f, Res(f,g)=∏ᵢ g(αᵢ) and it vanishes exactly when f and g have a common root Theorem

Dependency tree · two levels

5 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

J. S. Milne, Fields and Galois Theory, Proposition 4.35 through Example 4.37 (standard reference, not scraped)

The monic resultant Res⁡(f,g)Res(f,g) from the symmetric coefficient expression of ∏ig(xi)∏i​g(xi​)