DefinitionDefinition: AI-adaptedProof: Not applicableSession-authored (Fable 5 assisted)audited 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 · next 3 levels

Direct dependencies and their dependencies through the next three levels: 11 results over 5 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click 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​)