False statementConstruction: Literature-sourcedVerification: AI-adaptedprecheck passaudited 2026-08-26

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FALSE: every cyclotomic polynomial has all coefficients in ${- 1, 0, 1}$

Statement

False claim. Every coefficient of every cyclotomic polynomial $Φ_{n} \in Z [t]$ lies in ${- 1, 0, 1}$ .

The failure first appears at $n = 105$ : the coefficient of $t^{7}$ in $Φ_{105} (t)$ is $- 2$ .

Facts & Assumptions

Given: The cyclotomic recursion $\prod_{d ∣ n} Φ_{d} (t) = t^{n} - 1$ for $n \geq 1$ (The cyclotomic polynomials $Φ_{n} \in Z [t]$ , defined by $\prod_{d ∣ n} Φ_{d} = t^{n} - 1$ ).

[L1]

For every $n \geq 1$ , $Φ_{n}$ is a monic polynomial in $Z [t]$ and the displayed divisor recursion holds (The recursion defines a unique monic $Φ_{n} \in Z [t]$ , of degree $φ (n)$ ).

Refutation

technique · direct

1.1givenL1algebra

Running the divisor recursion for $n = 105 = 3 \cdot 5 \cdot 7$ and truncating modulo $t^{8}$ gives $Φ_{105} (t) \equiv \frac{(1 - t^{3}) (1 - t^{5}) (1 - t^{7})}{1 - t} \equiv 1 + t + t^{2} - t^{5} - t^{6} - 2 t^{7} (m o d t^{8}) .$ The first congruence is the defining recursion with every factor of degree at least $15$ dropped modulo $t^{8}$ , and the second comes from expanding $(1 - t)^{- 1} = 1 + t + t^{2} + t^{3} + t^{4} + t^{5} + t^{6} + t^{7} (m o d t^{8})$ .

2.1step 1.1algebra∎

Step 1.1 shows that the coefficient of $t^{7}$ in $Φ_{105} (t)$ is $- 2$ , and $- 2 \notin {- 1, 0, 1}$ . So the false claim fails.

Remarks

Why this is a real pattern and not a silly claim. For many small values of $n$ the coefficients do lie in ${- 1, 0, 1}$ , so the first counterexample is not visually obvious from the recursion alone.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

22 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), Section 5 (standard reference, not scraped)
P. L. Clark, Field Theory (course notes/monograph), Section 9.1.2 (standard reference, not scraped)

FALSE: every cyclotomic polynomial has all coefficients in {−1,0,1}{−1,0,1}