CorollaryStatement: AI-adaptedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-13

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Inside a common extension, the splitting field of $f g$ is the composite of the splitting fields of $f$ and $g$

Statement

Let $f, g \in F [x]$ be nonzero. Inside a common field extension $Ω / F$ , let $E_{f}$ and $E_{g}$ be their respective splitting fields. Then the composite $E_{f} E_{g}$ inside $Ω$ is a splitting field of $f g$ over $F$ .

Facts & Assumptions

Given: Nonzero $f, g \in F [x]$ and splitting fields $E_{f}, E_{g} \subseteq Ω$ .

[F1]

A splitting field is the subfield generated over $F$ by all roots of the polynomial, over which that polynomial splits (Polynomials that split and splitting fields of a polynomial or a family of polynomials).

[F2]

The composite $E_{f} E_{g}$ is the smallest subfield of $Ω$ containing both $E_{f}$ and $E_{g}$ (The composite of two subfields is the subfield generated by their union).

Proof

technique · direct

1.1

Both $f$ and $g$ split over $E_{f} E_{g}$ because the composite contains their splitting fields. Hence their product $f g$ splits there.

F1F2

2.1

In the field $Ω$ , an element $a$ is a root of $f g$ exactly when $f (a) g (a) = 0$ , hence exactly when it is a root of $f$ or of $g$ . Therefore the field generated by the roots of $f g$ is the field generated jointly by $E_{f}$ and $E_{g}$ , which is $E_{f} E_{g}$ by [F2].

F1F2algebra∎

Depends on

Used by

The splitting field of {x²-2,x²-3} over ℚ is ℚ(√2,√3) Example

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 11 results over 4 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, Chapter 2 (standard reference, not scraped)

Inside a common extension, the splitting field of fgfg is the composite of the splitting fields of ff and gg