ExampleConstruction: Literature-sourcedVerification: Literature-sourcedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-11

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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.
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Eisenstein proves $x^n-2$ irreducible over $\mathbb Q$ for every positive $n$

Example

For every positive natural number $n$ , the polynomial $x^n-2$ is irreducible in $\mathbb Q[x]$ .

Facts & Assumptions

Given: A natural number $n\ge1$ .

[L1]

For every prime $p$ and positive $n$ , the polynomial $x^n-p$ is irreducible over $\mathbb Q$ (For every prime $p$ and positive $n$ , $x^n-p$ is irreducible over $\mathbb Q$ ).

[L2]

An integer is prime when it exceeds $1$ and has no positive divisors other than $1$ and itself (Prime and composite integers: $p$ is prime when $p > 1$ and its only positive divisors are $1$ and $p$ ).

[L3]

A divisor of a nonzero integer is bounded in absolute value (If $d \mid a$ and $a \ne 0$ then $d \ne 0$ and $|d| \le |a|$ ; hence the set of divisors of a nonzero integer is bounded above by $|a|$ ).

[L4]

Integer absolute value is defined by sign cases (The absolute value $|a|$ of an integer).

[L5]

Integer absolute value is nonnegative and detects zero (Absolute value in $\mathbb{Z}$ : $|a| \ge 0$ ; $|a| = 0$ exactly when $a = 0$ ; $|-a| = |a|$ ; $|ab| = |a|\,|b|$ ; $-|a| \le a \le |a|$ ; and $|a| \le c$ exactly when $-c \le a \le c$ ).

[L6]

The integers form an ordered ring (The integers form a totally ordered ring).

[L7]

Natural numbers embed in the integers preserving arithmetic (The naturals embed in the integers).

[L8]

The natural order is discrete (Discreteness: $\sigma(n)$ is the immediate successor).

Verification

technique · direct

1.1

Facts [L2] through [L8] verify that $2>1$ and that its only positive divisors are $1$ and $2$ , so $2$ is prime.

givenL2L3L4L5L6L7L8

2.1

Apply [L1] with $p=2$ and the given positive $n$ to obtain the claimed irreducibility.

step 1.1L1∎

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: 64 results over 24 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

Keith Conrad, Irreducibility Tests in Q[T], Example 1.6 (standard reference, not scraped)

Eisenstein proves xn−2x^n-2xn−2 irreducible over Q\mathbb QQ for every positive nnn