ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passaudited 2026-08-16

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.

$Q (\sqrt[3]{2}) / Q$ has degree three

Example

If $a = \sqrt[3]{2}$ , then $Q (a) / Q$ has degree $3$ and basis $(1, a, a^{2})$ .

Facts & Assumptions

Given: The positive real cube root $a$ of $2$ .

[L1]

A simple algebraic extension has degree equal to the degree of the element's minimal polynomial and has the corresponding power basis (A simple algebraic extension is its minimal-polynomial quotient and has power basis $1, a, \dots, a^{n - 1}$ and degree $n$ ).

[L2]

Eisenstein's criterion proves suitable primitive integer polynomials irreducible over $Q$ (Eisenstein criterion over the integers).

[L3]

The positive real $n$ -th root exists uniquely for every positive input and $n \geq 1$ (Existence and uniqueness of $n$ -th roots: a unique $a^{1 / n} \geq 0$ with $(a^{1 / n})^{n} = a$ ).

Verification

technique · direct

1.1givenL2L3

By [L3], $a^{3} = 2$ . The polynomial $t^{3} - 2$ is Eisenstein at $2$ , so it is irreducible by [L2].

2.1step 1.1L1∎

Its minimal polynomial therefore has degree $3$ , and [L1] gives degree $3$ and basis $(1, a, a^{2})$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

30 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

A. W. Knapp, Basic Algebra, 2nd ed., Chapter IX, Section 1 (standard reference, not scraped)

Q(23)/QQ(32​)/Q has degree three