ExampleConstruction: AI-adaptedVerification: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + claude-sonnet-5)audited 2026-08-17

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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.

$Q (\sqrt{2}, \sqrt{3})$ has four embeddings into $\overline{Q}$

Example

The field $E = Q (\sqrt{2}, \sqrt{3})$ has degree four over $Q$ and has four embeddings into an algebraic closure. They are the independent sign choices

$(\sqrt{2}, \sqrt{3}) ⟼ (\pm \sqrt{2}, \pm \sqrt{3}) .$

Facts & Assumptions

Given: Positive real square roots $\sqrt{2}, \sqrt{3}$ and an algebraic closure $Ω / Q$ .

[L1]

Eisenstein's criterion proves the irreducibility of the integer polynomials used below (Eisenstein criterion over the integers).

[L2]

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

[L3]

Ordinary degrees multiply in finite towers (Tower law for finite extensions: $[L : F] = [L : K] [K : F]$ ).

[L4]

Embeddings of a simple extension correspond to distinct roots of its minimal polynomial ( $F$ -embeddings of $F (α)$ into an algebraically closed field correspond to the distinct roots of $m_{α}$ ).

[L5]

Restriction partitions embeddings in a finite tower into equal extension fibres (Restriction partitions embeddings in a finite tower into extension fibres).

[L6]

The notation $F (α, β)$ denotes the subfield generated by the named elements (Field extensions, generated subrings $F [S]$ , generated subfields $F (S)$ , and simple extensions).

Verification

technique · direct

1.1L1L2

Eisenstein at $2$ , $3$ , and $2$ makes $x^{2} - 2$ , $x^{2} - 3$ , and $x^{2} - 6$ irreducible over $Q$ by [L1]. Thus none of $\sqrt{2}, \sqrt{3}, \sqrt{6}$ is rational, and [L2] gives the basis $(1, \sqrt{2})$ of $Q (\sqrt{2}) / Q$ .

2.1step 1.1L2algebra

If $\sqrt{3} = a + b \sqrt{2}$ with $a, b \in Q$ , squaring and using the basis gives $2 a b = 0$ . If $b = 0$ , then $\sqrt{3}$ is rational; if $a = 0$ , then $2 b$ is a rational square root of $6$ . Both contradict step 1.1. Hence $x^{2} - 3$ has no root in $Q (\sqrt{2})$ and is irreducible there.

3.1step 1.1step 2.1L2L3

By [L2] the second tower step has degree two, and [L3] gives $[E : Q] = 4$ .

4.1L4L5L6∎

The first square root has two distinct images by [L4]. Over each image of $Q (\sqrt{2})$ , the second square root has the two distinct images $\pm \sqrt{3}$ ; [L5] shows these fibres exhaust all extensions. Thus the four independent sign choices are exactly the four embeddings.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

28 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

P. L. Clark, Field Theory, Chapter 4 (standard reference, not scraped)
J. S. Milne, Fields and Galois Theory, Chapters 2 to 6 (standard reference, not scraped)

Q(2,3)Q(2​,3​) has four embeddings into Q‾Q​