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

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Translation turns $x^4+1$ into an Eisenstein polynomial

Example

The polynomial $x^4+1$ is irreducible over $\mathbb Q$ , although Eisenstein's criterion does not apply to it before a translation.

Facts & Assumptions

Given: The polynomial $f=x^4+1\in\mathbb Q[x]$ and the substitution $\tau(f)(x)=f(x+1)$ .

[L1]

The universal property makes substitution $x\mapsto x+1$ a ring homomorphism; substitution $x\mapsto x-1$ is its inverse (Universal property of $R[x]$ : a coefficient homomorphism and the image of $x$ determine a unique ring homomorphism).

[L2]

Eisenstein's criterion proves a primitive integer polynomial irreducible over $\mathbb Q$ from the stated prime-divisibility conditions (Eisenstein criterion over the integers).

[L3]

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 $p$ ).

[L4]

The rational numbers form a field (The rationals form a field).

Verification

technique · direct

1.1

No prime can make $f=x^4+1$ Eisenstein before translation, because the criterion would require that prime to divide the constant coefficient $1$ . A direct divisor check using [L3] shows that $2$ is prime. Binomial expansion gives $\tau(f)=x^4+4x^3+6x^2+4x+2$ . It is monic and hence primitive, and it is Eisenstein at $2$ : every nonleading coefficient is even, the leading coefficient is odd, and $4$ does not divide the constant coefficient $2$ . Thus [L2] makes $\tau(f)$ irreducible.

givenL2L3algebra

2.1

Over the field [L4], [L1] makes $\tau$ an automorphism; a nontrivial factorization of $f$ would map to one of $\tau(f)$ , so irreducibility of $\tau(f)$ implies irreducibility of $f$ .

step 1.1L1L4∎

Depends on

Used by

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Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 88 results over 22 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

Neil Donaldson, Math 120B Notes, translated Eisenstein example (standard reference, not scraped)

Translation turns x4+1x^4+1x4+1 into an Eisenstein polynomial