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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.
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Eisenstein proves $x^{n} - 2$ irreducible over $Q$ for every positive $n$

Example

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

Facts & Assumptions

Given: A natural number $n \geq 1$ .

[L1]

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

[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 $d ∣ a$ and $a \neq 0$ then $d \neq 0$ and $∣ d ∣ \leq ∣ a ∣$ ; 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 $Z$ : $∣ a ∣ \geq 0$ ; $∣ a ∣ = 0$ exactly when $a = 0$ ; $∣ - a ∣ = ∣ a ∣$ ; $∣ a b ∣ = ∣ a ∣ ∣ b ∣$ ; $- ∣ a ∣ \leq a \leq ∣ a ∣$ ; and $∣ a ∣ \leq c$ exactly when $- c \leq a \leq c$ ).

[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: $σ (n)$ 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 · two levels

35 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

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

Eisenstein proves xn−2xn−2 irreducible over QQ for every positive nn