ExampleConstruction: Literature-sourcedVerification: AI-adaptedverified 2026-09-23 (gpt-6-sol)

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The two nontrivial characters of $C_{3}$ form one rational representation

Example

For $C_{3} = ⟨ g ∣ g^{3} = 1 ⟩$ and $ζ = e^{2 π i / 3}$ , the characters $χ (g) = ζ$ and $\overline{χ} (g) = ζ^{2}$ are Galois conjugate. Their sum is afforded over $Q$ by the action of $g$ on $Q^{2}$ with matrix $(\begin{array}{cc} 0 & - 1 \\ 1 & - 1 \end{array}) .$ Thus the rational Galois orbit occurs with multiplicity one. Each of the two characters also has an explicit one-dimensional model over its character field $Q (ζ)$ , with scalar-extension multiplicity one. Under the Axiom of Choice, the general Schur-index definition therefore gives $m_{Q (ζ)} (χ) = m_{Q (ζ)} (\overline{χ}) = 1$ .

Facts & Assumptions

Given: $C_{3}$ , $g$ , and $ζ$ as displayed.

[L1]

Over a nonsplitting field, an irreducible character orbit occurs with a common scalar-extension multiplicity (Character formula over a nonsplitting field).

[L2]

Under AC, the Schur index is the multiplicity attached to the unique irreducible representation over the character field (The Schur index of an irreducible character).

Verification

technique · computation

1.1algebra

The displayed matrix has characteristic polynomial $x^{2} + x + 1$ , hence eigenvalues $ζ, ζ^{2}$ , and its cube is the identity. It therefore gives a rational $C_{3}$ -representation with complex character $χ + \overline{χ}$ . Since $x^{2} + x + 1$ has no rational root, this representation has no invariant rational line and is irreducible over $Q$ .

2.1L1L2step 1.1algebra∎

The orbit appears once, so [L1] identifies its scalar-extension multiplicity over $Q$ as $1$ . Over $K = Q (ζ)$ , the maps $g \mapsto ζ$ and $g \mapsto ζ^{2}$ each define an irreducible one-dimensional $K$ -representation whose complex scalar extension has multiplicity one. These explicit computations use no choice principle. Under AC, [L2] identifies each attached representation and multiplicity with its Schur index over $K$ , giving the two asserted values.

Depends on

Used by

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Sources

Gabor Wiese, Galois Representations, Section 2.5 (standard reference, not scraped)

The two nontrivial characters of C3C3​ form one rational representation