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Total ramification at a prime-power cyclotomic level

Statement

For a prime $p$ and an integer $a \geq 1$ , put $K = Q (ζ_{p^{a}})$ and $λ = 1 - ζ_{p^{a}}$ . Then $p O_{K} = (λ)^{φ (p^{a})},$ and the principal ideal $λ O_{K}$ is the unique prime ideal of $O_{K}$ lying above $p$ , with residue field $O_{K} / λ O_{K} ≅ F_{p}$ .

Facts & Assumptions

Given: A prime $p$ , an integer $a \geq 1$ , $e : = φ (p^{a})$ , a primitive $p^{a}$ -th root of unity $ζ = ζ_{p^{a}}$ , $K : = Q (ζ)$ , $λ : = 1 - ζ$ and $R : = Z [ζ] = O_{K}$ .

[F1]

$O_{K} = R = Z [ζ]$ and $p R = (λ)^{e}$ ; moreover $1, ζ, \dots, ζ^{e - 1}$ is an integral basis of $O_{K}$ (Prime-power cyclotomic ring, discriminant support and p factor, Ring of integers of every cyclotomic field).

[F2]

$Φ_{p^{a}} \in Z [t]$ is monic of degree $e$ and $Φ_{p^{a}} (1) = p$ ( $Φ_{p^{r}} (t) = \sum_{k < p} t^{k p^{r - 1}}$ , and $Φ_{p^{r}} (t + 1)$ is Eisenstein at $p$ ).

[F3]

Division by the monic polynomial $t - 1$ in $Z [t]$ writes every $f \in Z [t]$ uniquely as $f = q \cdot (t - 1) + f (1)$ with $q \in Z [t]$ and constant remainder $f (1)$ (Division by a monic polynomial over a commutative ring); hence the evaluation homomorphism $Z [t] \to Z$ , $f \mapsto f (1)$ , is a surjective ring homomorphism with kernel $(t - 1)$ , so $Z [t] / (t - 1) ≅ Z$ (First isomorphism theorem for rings: $R / \ker f ≅ im f$ ).

[F4]

An ideal $M$ of a commutative ring $R$ is maximal if and only if $R / M$ is a field ( $R / M$ is a field if and only if $M$ is a maximal ideal), and every maximal ideal is prime (Every maximal ideal of a commutative ring is prime). Also $Z / p$ is a field (For every prime $p$ , the two operations on $Z / p$ make it a field).

[F5]

A nonzero prime $P$ of $O_{K}$ lies above $p$ when $P \cap Z = p Z$ , and its residue degree is $[O_{K} / P : F_{p}]$ (Primes above and residue degree).

Proof

technique · direct

1.1F1F2F3F4

Under the presentation $R = Z [ζ] = Z [t] / (Φ_{p^{a}})$ with $t \mapsto ζ$ , the element $λ = 1 - ζ$ corresponds to the class of $1 - t$ , so $R / λ R ≅ Z [t] / (Φ_{p^{a}}, t - 1)$ ; sending $t$ to $1$ via [F3] identifies this quotient with $Z / (Φ_{p^{a}} (1)) = Z / p Z = F_{p}$ .

2.1F1F4F5step 1.1

Since $R / λ R ≅ F_{p}$ is a field, $λ R$ is a maximal and hence prime ideal of $R$ , and it is proper; consequently $λ R \cap Z$ is a proper ideal of $Z$ containing $p Z$ (as $p R = (λ)^{e} \subseteq λ R$ by [F1]) and therefore equals $p Z$ , so $λ R$ lies above $p$ with residue field $O_{K} / λ O_{K} ≅ F_{p}$ , of residue degree $1$ .

3.1F1F4step 2.1

If $P$ is any prime ideal of $R$ with $p \in P$ , then $λ^{e} \in λ^{e} R = p R \subseteq P$ , so $λ \in P$ because $P$ is prime, hence $λ R \subseteq P$ ; as $λ R$ is maximal and $P$ is proper, $P = λ R$ . Thus $λ R$ is the unique prime above $p$ .

4.1F1step 2.1step 3.1∎

Combining [F1] with steps 2.1 and 3.1, $p O_{K} = (λ)^{φ (p^{a})}$ and $λ O_{K}$ is the unique prime of $O_{K}$ above $p$ , with residue field $F_{p}$ . For $p = 2$ , $a = 1$ this reads $K = Q$ , $λ = 2$ , $e = 1$ , and $2 Z$ is the unique prime above $2$ with residue field $F_{2}$ .

Remarks

Total ramification. The exponent equals the degree $[Q (ζ_{p^{a}}) : Q] = φ (p^{a})$ , and the residue degree is $1$ , so $p$ is totally ramified; the equality $p O_{K} = (λ)^{e}$ exhibits the ramification index without invoking any general ramification theory beyond the definitions.
The quotient computation is the only place where $Φ_{p^{a}} (1) = p$ is used, and it also shows that no prime other than $λ O_{K}$ can contain $p$ .

Depends on

Used by

Arithmetic of Q(zeta₅) Example
Prime factorisation in a cyclotomic field Theorem

Dependency tree · two levels

71 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

J. S. Milne, Algebraic Number Theory, Proposition 6.2(c) and proof, pp. 96-97 (standard reference, not scraped)
Conrad-Landesman, Math 154 Algebraic Number Theory, Corollary 10.7 (standard reference, not scraped)