Representation Theory of Finite Groups by Benjamin Steinberg

Author:Benjamin Steinberg
Language: eng
Format: epub
Publisher: Springer New York, New York, NY

Then α is an algebraic integer. By Corollary 4.1.10, ϕg is diagonalizable and its eigenvalues λ1, …, λd are nth-roots of unity. Since ϕg is diagonalizable but not a scalar matrix, its eigenvalues are not all the same. Applying Lemma 6.3.1 to yields | χ(g) |  < d, and so

Also note that . Let σ ∈ Γ. Lemma 6.3.6 implies that σ(α) is an algebraic integer. Corollary 6.3.4 tells us that

is again a sum of dnth-roots of unity, not all equal. Hence, another application of Lemma 6.3.1 yields

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