CorollaryStatement: Literature-sourcedProof: Literature-sourcedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-11

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For every prime $p$ and positive $n$ , $x^n-p$ is irreducible over $\mathbb Q$

Statement

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

Facts & Assumptions

Given: A prime integer $p$ and a natural number $n\ge1$ .

[L1]

A primitive integer polynomial is irreducible over $\mathbb Q$ when a prime divides every nonleading coefficient, does not divide the leading coefficient, and its square does not divide the constant coefficient (Eisenstein criterion over the integers).

[L2]

A prime integer satisfies $p>1$ and has no positive divisor other than $1,p$ (Prime and composite integers: $p$ is prime when $p > 1$ and its only positive divisors are $1$ and $p$ ).

[L3]

Nonzero integers cancel in products (The integers have no zero divisors; multiplicative cancellation).

[L4]

The only integer units are $1$ and $-1$ ( $(\mathbb{Z}, \cdot, 1)$ is a commutative monoid whose group of units is $\{1, -1\}$ ; equivalently $u \mid 1$ holds exactly for $u = 1$ and $u = -1$ ).

Proof

technique · direct

1.1

The polynomial $x^n-p$ is primitive because its leading coefficient is the unit $1$ from [L4]; the prime $p$ divides every nonleading coefficient, including the zero intermediate coefficients, and does not divide $1$ .

givenL2L4

2.1

If $p^2$ divided $p$ , cancellation by the nonzero $p$ using [L3] would make $p$ a unit, contradicting [L2] and [L4]; thus Eisenstein's criterion [L1] applies and proves irreducibility.

step 1.1L1L2L3L4∎

Depends on

Used by

Eisenstein proves xⁿ-2 irreducible over ℚ for every positive n Example

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 84 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)

For every prime ppp and positive nnn, xn−px^n-pxn−p is irreducible over Q\mathbb QQ