ExampleConstruction: Literature-sourcedVerification: Literature-sourcedprecheck passverified 2026-09-23 (gpt-6-sol)

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The trivial character has Schur index one

Example

For every finite group $G$ , the trivial character $1_{G}$ has character field $Q$ and an explicit one-dimensional rational model with scalar-extension multiplicity one. Under the Axiom of Choice, its Schur index is $m_{Q} (1_{G}) = 1$ .

Facts & Assumptions

Given: A finite group $G$ and its one-dimensional trivial complex representation.

[L1]

Under AC, the Schur index is the least positive multiplicity with a model over the character field (Schur index as minimal realization multiplicity).

Verification

technique · direct

1.1L1algebra

Every value of $1_{G}$ is $1$ , so its character field is $Q$ . The one-dimensional representation on $Q$ in which every element acts as $1$ is a $Q$ -model.

2.1L1step 1.1∎

The rational one-dimensional model has scalar-extension multiplicity one by direct inspection, without any choice principle. Under AC, [L1] makes the Schur index $1$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

15 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

Gabor Wiese, Galois Representations, Corollary 2.5.10 (standard reference, not scraped)