ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck 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 $Q$ , although Eisenstein's criterion does not apply to it before a translation.

Facts & Assumptions

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

[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 $Q$ 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 $τ (f) = x^{4} + 4 x^{3} + 6 x^{2} + 4 x + 2$ . 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 $τ (f)$ irreducible.

givenL2L3algebra

2.1

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

step 1.1L1L4∎

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Sources

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

Translation turns x4+1x4+1 into an Eisenstein polynomial