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Arithmetic Frobenius is the power map in an unramified cyclotomic field

Statement

Let $f \geq 1$ be a reduced index, that is $f$ is odd or $4 ∣ f$ , let $ℓ$ be a rational prime with $ℓ ∤ f$ , let $ζ_{f}$ be a primitive $f$ -th root of unity and $K = Q (ζ_{f})$ . Let $σ_{ℓ}$ be the automorphism of $K$ with $σ_{ℓ} (ζ_{f}) = ζ_{f}^{ℓ}$ . Then for every prime $P$ of $O_{K}$ above $ℓ$ , the element $σ_{ℓ}$ is the arithmetic Frobenius ${Frob}_{P}$ : it is the unique $σ \in Gal (K / Q)$ with $σ (P) = P$ and $σ (a) \equiv a^{ℓ} (m o d P)$ for all $a \in O_{K}$ . In particular it does not depend on the chosen prime above $ℓ$ , the Galois group being abelian.

Facts & Assumptions

Given: A reduced index $f \geq 1$ , a rational prime $ℓ$ with $ℓ ∤ f$ , a primitive $f$ -th root of unity $ζ = ζ_{f}$ , the field $K = Q (ζ)$ , and the automorphism $σ_{ℓ} \in Gal (K / Q)$ with $σ_{ℓ} (ζ) = ζ^{ℓ}$ .

[F1]

$O_{K} = Z [ζ]$ , and $1, ζ, \dots, ζ^{φ (f) - 1}$ is an integral basis (Ring of integers of every cyclotomic field).

[F2]

$Φ_{f}$ is the monic minimal polynomial of $ζ$ over $Q$ and has degree $φ (f)$ ( $Φ_{n}$ is irreducible in $Q [t]$ for every $n \geq 1$ , The recursion defines a unique monic $Φ_{n} \in Z [t]$ , of degree $φ (n)$ ).

[F3]

$K / Q$ is Galois with $Gal (K / Q) ≅ (Z / f)^{\times}$ via $σ_{b} (ζ) = ζ^{b}$ ; this group is abelian and every automorphism has this form ( $[Q (ζ_{n}) : Q] = φ (n)$ and $Gal (Q (μ_{n}) / Q) ≅ (Z / n)^{\times}$ ).

[F4]

For $\gcd (f, ℓ) = 1$ , the reduction of $Φ_{f}$ in $F_{ℓ} [t]$ is a product of pairwise distinct monic irreducibles, each of degree $d : = {ord}_{f} (ℓ)$ (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 $n$ ).

[F5]

Since $O_{K} = Z [ζ]$ , applying the monogenic factorisation lemma to ${\overset{ˉ}{Φ}}_{f} = \prod_{i} g_{i}$ gives $ℓ O_{K} = \prod_{i} P_{i}$ where $P_{i} = (ℓ, {\tilde{g}}_{i} (ζ))$ are distinct primes above $ℓ$ , each of residue degree $d$ , where ${\tilde{g}}_{i} \in Z [t]$ is the coefficientwise lift of $g_{i}$ with coefficients in ${0, \dots, ℓ - 1}$ (Choice-free prime factorisation for a monogenic number ring, Primes above and residue degree).

[F6]

Over a field whose characteristic does not divide $f$ , the roots of $Φ_{f}$ in a splitting field of $t^{f} - 1$ are exactly the primitive $f$ -th roots of unity. For the residue field $κ (P_{i})$ , take a splitting field of $t^{f} - 1$ over it; its natural field embedding is injective and preserves the multiplicative order of each element. Since $κ (P_{i})$ has characteristic $ℓ$ and $ℓ ∤ f$ , the theorem applies to the image of $\overset{ˉ}{ζ}$ there (Over a field whose characteristic does not divide $n$ , the roots of $Φ_{n}$ are exactly the primitive roots of unity, The cyclotomic extension $K (μ_{n})$ as a splitting field of $t^{n} - 1$ ).

[F7]

For a prime $P$ above an unramified rational prime $ℓ$ in a finite Galois extension, there is a unique arithmetic Frobenius ${Frob}_{P}$ in the decomposition group satisfying ${Frob}_{P} (a) \equiv a^{ℓ} (m o d P)$ for all algebraic integers $a$ (Unramified frobenius element exists uniquely).

Proof

technique · direct

1.1F1F2F3

By [F1] and [F2], the ring of integers is $Z [ζ]$ , the minimal polynomial of $ζ$ is $Φ_{f}$ , and its degree is $φ (f)$ . By [F3], $K / Q$ is abelian and each automorphism is $σ_{b}$ for a unit class $b$ modulo $f$ , in particular $σ_{ℓ}$ exists. If $f = 1$ , then $K = Q$ and the unique prime over $ℓ$ is $ℓ Z$ ; the trivial automorphism is its arithmetic Frobenius.

1.2F4F5

Since $ℓ ∤ f$ , [F4] and [F5] give $ℓ O_{K} = \prod_{i = 1}^{g} P_{i}$ with distinct primes $P_{i} = (ℓ, {\tilde{g}}_{i} (ζ))$ , each of residue degree $d = {ord}_{f} (ℓ)$ . Thus $ℓ$ is unramified and these are all the primes above it.

1.3F7

Fix $i$ . By [F7] the prime $P_{i}$ has an arithmetic Frobenius ${Frob}_{P_{i}}$ satisfying the $ℓ$ -power congruence. Evaluating it at $ζ$ gives ${Frob}_{P_{i}} (ζ) \equiv ζ^{ℓ} (m o d P_{i}) .$

1.4F6

The residue class $\overset{ˉ}{ζ} = ζ + P_{i}$ is a root of the reduction of $Φ_{f}$ , since $Φ_{f} (ζ) = 0$ . Embed the residue field into a splitting field of $t^{f} - 1$ over it. By [F6], the image of $\overset{ˉ}{ζ}$ has multiplicative order exactly $f$ there; injectivity of the field embedding gives the same order for $\overset{ˉ}{ζ}$ in the residue field.

2.1F3step 1.3step 1.4

Write ${Frob}_{P_{i}} = σ_{b}$ using [F3]. Reducing the congruence in step 1.3 gives ${\overset{ˉ}{ζ}}^{b} = {\overset{ˉ}{ζ}}^{ℓ} .$ Since $\overset{ˉ}{ζ}$ has order $f$ by step 1.4, $b \equiv ℓ (m o d f)$ ; hence $σ_{b} = σ_{ℓ}$ . This comparison is made directly in the residue field at $P_{i}$ , so it does not require $σ_{ℓ}$ to stabilise $P_{i}$ in advance.

3.1F3step 2.1∎

The argument applies to every prime $P_{i}$ above $ℓ$ , and each gives the same automorphism $σ_{ℓ}$ . Thus this arithmetic Frobenius is independent of the prime above $ℓ$ , as also follows from the abelian Galois group in [F3].

Remarks

Reduced index. The hypothesis that $f$ is odd or $4 ∣ f$ is the standing reduced-index convention of this pair; for the present lemma the essential hypothesis is $ℓ ∤ f$ , which makes $Φ_{f}$ separable modulo $ℓ$ .
Power map, not inverse. The identification uses the arithmetic convention $ζ \mapsto ζ^{ℓ}$ ; the geometric inverse would send $ζ$ to $ζ^{ℓ^{- 1}}$ and agrees with the arithmetic map exactly when $ℓ^{2} \equiv 1 (m o d f)$ , since $ℓ$ is a unit modulo $f$ .

Depends on

Used by

Complete splitting criterion for a cyclotomic field Corollary
Decomposition of an unramified prime in a cyclotomic field Corollary
First supplement from Frobenius on Q(i) Corollary
Second supplement from Frobenius on Q(zeta₈) Corollary
Second supplement in four residue classes modulo eight Example
Quadratic reciprocity as a Frobenius restriction identity Theorem

Dependency tree · two levels

73 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, Ch. 8, Example 8.18 and Ch. 6 Remark 6.6 (standard reference, not scraped)
Conrad-Landesman, Math 154 Algebraic Number Theory, Chs. 11-12 (standard reference, not scraped)