False statementConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-26

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FALSE: $Φ_{n}$ is irreducible over every field

Statement

False claim. For every field $K$ and every $n \geq 1$ with the characteristic of $K$ not dividing $n$ , the image of $Φ_{n}$ in $K [t]$ is irreducible.

What is true is different in two directions: over $Q$ every $Φ_{n}$ is irreducible, but over a finite field the factor degree is governed by the order of the Frobenius class modulo $n$ .

Facts & Assumptions

Given: The rational irreducibility theorem and the finite-field factorisation theorem.

[L1]

For every $n \geq 1$ , the cyclotomic polynomial $Φ_{n}$ is irreducible in $Q [t]$ ( $Φ_{n}$ is irreducible in $Q [t]$ for every $n \geq 1$ ).

[L2]

If $\gcd (n, q) = 1$ , the reduction of $Φ_{n}$ in $F_{q} [t]$ is a product of distinct monic irreducibles, each of degree the order of $[q]$ modulo $n$ (For $\gcd (n, q) = 1$ the reduction of $Φ_{n}$ in $F_{q} [t]$ is a product of distinct monic irreducibles, each of degree the order of $[q]$ modulo $n$ ).

Refutation

technique · direct

1.1

In $(Z / 5)^{\times}$ one has $[11] = [1]$ , so the order of $[11]$ modulo $5$ is $1$ . Therefore [L2] says that over $F_{11}$ every irreducible factor of $Φ_{5}$ has degree $1$ .

L2algebra

2.1

Hence the reduction of $Φ_{5}$ in $F_{11} [t]$ is a product of distinct linear factors, so it is reducible there. This contradicts the false claim.

step 1.1algebra

3.1

The contradiction does not touch [L1]: irreducibility over $Q$ is a theorem, but it does not persist over arbitrary base fields.

step 2.1L1∎

Remarks

The finite-field theorem is the correct replacement. The question over $F_{q}$ is not "irreducible or not?" in the abstract, but "what is the order of $[q]$ modulo $n$ ?"

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

32 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

K. Conrad, Cyclotomic Extensions (expository blurb), Theorem 5.4 and Corollary 5.7 (standard reference, not scraped)
P. L. Clark, Field Theory (course notes/monograph), Theorem 9.8 (standard reference, not scraped)

FALSE: ΦnΦn​ is irreducible over every field