CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-13

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The composite of two subfields is the subfield generated by their union

Statement

If $E$ and $E^{'}$ are subfields of a field $Ω$ , then $E (E^{'}) = E^{'} (E) = ⋂ {L : L is a subfield of Ω, E \cup E^{'} \subseteq L} .$ This common subfield is denoted $E E^{'}$ and is called the composite of $E$ and $E^{'}$ .

Facts & Assumptions

Given: Subfields $E, E^{'} \subseteq Ω$ .

[F1]

For a subfield $F \subseteq K$ and $S \subseteq K$ , $F (S)$ is the smallest subfield of $K$ containing $F \cup S$ (Field extensions, generated subrings $F [S]$ , generated subfields $F (S)$ , and simple extensions).

Proof

technique · direct

1.1

A subfield $L \subseteq Ω$ contains $E \cup E^{'}$ if and only if it contains $E$ and every element of $E^{'}$ .

algebra

2.1

By minimality, $E (E^{'})$ is the intersection of precisely the subfields described in step 1.1.

F1step 1.1

2.2

Interchanging $E$ and $E^{'}$ shows that $E^{'} (E)$ is the same intersection.

F1step 1.1

3.1

Hence both generated fields equal the displayed intersection; denote it by $E E^{'}$ .

step 2.1step 2.2∎

Depends on

Field extensions, generated subrings $F[S]$, generated subfields $F(S)$, and simple extensions

Used by

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 (standard reference, not scraped)