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$Φ_{7} (t + 1)$ is Eisenstein at seven

Example

The translated seventh cyclotomic polynomial is

$Φ_{7} (t + 1) = t^{6} + 7 t^{5} + 21 t^{4} + 35 t^{3} + 35 t^{2} + 21 t + 7.$

Its leading coefficient is $1$ , every other coefficient is divisible by $7$ , and its constant term $7$ is not divisible by $7^{2}$ . So it satisfies Eisenstein's criterion at the prime $7$ , and therefore $Φ_{7}$ is irreducible over $Q$ .

Facts & Assumptions

Given: The prime-power cyclotomic formula and the polynomial $Φ_{7}$ .

[L1]

For a prime $p$ and $r \geq 1$ , $Φ_{p^{r}} (t) = \sum_{k = 0}^{p - 1} t^{k p^{r - 1}},$ and $Φ_{p^{r}} (t + 1)$ is Eisenstein at $p$ ( $Φ_{p^{r}} (t) = \sum_{k < p} t^{k p^{r - 1}}$ , and $Φ_{p^{r}} (t + 1)$ is Eisenstein at $p$ ).

[L2]

Eisenstein's criterion: if a prime $p$ divides every non-leading coefficient of a polynomial in $Z [t]$ , does not divide the leading coefficient, and $p^{2}$ does not divide the constant term, then the polynomial is irreducible over $Q$ (Eisenstein criterion over the integers).

Verification

technique · direct

1.1

Applying [L1] at $p = 7$ and $r = 1$ gives $Φ_{7} (t) = 1 + t + t^{2} + t^{3} + t^{4} + t^{5} + t^{6}$ .

2.1

Therefore $Φ_{7} (t + 1) = \frac{(t + 1)^{7} - 1}{t} = t^{6} + 7 t^{5} + 21 t^{4} + 35 t^{3} + 35 t^{2} + 21 t + 7,$ by the binomial theorem.

step 1.1algebra

3.1

In the polynomial of step 2.1 the leading coefficient is $1$ , the remaining coefficients $7, 21, 35, 35, 21, 7$ are all divisible by $7$ , and the constant term $7$ is not divisible by $49$ ; so [L2] applies at the prime $7$ .

step 2.1L2algebra

4.1

Hence $Φ_{7} (t + 1)$ is irreducible over $Q$ , and this is exactly the degree-one prime-power case of [L1].

step 3.1L1∎

Remarks

Why this example matters later. The explicit coefficients are what the counterexample page uses when it says the Eisenstein route already proves irreducibility for prime-power cyclotomic polynomials before the general Dedekind argument is built.

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Sources

K. Conrad, Cyclotomic Extensions (expository blurb), Example 2.2 (standard reference, not scraped)
J. S. Milne, Fields and Galois Theory, v5.10, Lemma 1.42 (standard reference, not scraped)

Φ7(t+1)Φ7​(t+1) is Eisenstein at seven