LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05

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The Teichmuller lift is multiplicative and unique

Statement

In a fixed splitting $p$ -modular system, each element of $k^{\times}$ of order prime to $p$ has exactly one Teichmuller lift, and the lift satisfies

$τ (λ μ) = τ (λ) τ (μ)$

whenever $λ, μ \in k^{\times}$ have order prime to $p$ .

Facts & Assumptions

Given: A splitting $p$ -modular system $(K, O, k)$ and $λ, μ \in k^{\times}$ of order prime to $p$ .

[F1]

The Teichmuller lift of such an element is, by definition, the unique prime-to- $p$ root of unity in $O^{\times}$ reducing to it (Teichmuller lift in a splitting p-modular system).

Proof

technique · direct

1.1

The uniqueness clause is already built into [F1]: if $u, v \in O^{\times}$ both have prime-to- $p$ order and reduce to $λ$ , then both satisfy the defining property of the Teichmuller lift of $λ$ , so $u = v$ .

F1given

2.1

The product $τ (λ) τ (μ)$ reduces to $λ μ$ in $k^{\times}$ . It is again a root of unity of order prime to $p$ , because it lies in the finite subgroup generated by two prime-to- $p$ roots of unity. By the uniqueness from step 1.1, it must equal the Teichmuller lift of $λ μ$ .

F1step 1.1algebra

3.1

Steps 1.1 and 2.1 give the claimed uniqueness and multiplicativity.

step 1.1step 2.1∎

Depends on

Teichmuller lift in a splitting p-modular system

Used by

The Brauer character is independent of basis and splitting-field realization Lemma

Dependency tree · two levels

2 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. Miquel Martinez, Modular Representation Theory of Finite Groups (standard reference, not scraped)
Tudor Ciurca, Representation Theory (standard reference, not scraped)