ExampleConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck 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.

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$Q (\sqrt{2}, \sqrt{3})$ 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

1.1

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 $Q (\sqrt{2}) / Q$ .

L1L2

2.1

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.

step 1.1L2algebra

3.1

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

step 1.1step 2.1L2L3

4.1

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.

L4L5L6∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 99 results over 16 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

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​