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Proof in characteristic zero
| 3 || <math>\alpha</math> is the sum <math>\sum_{i=1}^r d_i\chi_i</math> || Facts (1),(2) || <math>\chi_i</math> are characters of (all the) irreducible representations. || Step (2) || <toggledisplay>We know by Fact (1) that <matH>\alpha</math> is a nonnegative integer combination of the <math>\chi_i</math>s, because <math>\rho</math> is a nonnegative integer combination of the corresponding representations. By Fact (2) and Step (2), the coefficients are precisely <math>\langle \alpha, \chi_i \rangle = d_i</math>.</toggledisplay>
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| 4 || The value of <math>\alpha</math> iat at the identity element is <math>\sum_{i=1}^r d_i^2</math>. || || || Step (3) || <toggledisplay>At the identity element, <math>\chi_i</math> takes the value <math>d_i</math>, so plugging in Step (3) gives this.</toggledisplay>
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| 5 || <math>\sum_{i=1}^r d_i^2 = |G|</math> || || || Steps (1), (4) || <toggledisplay>By Step (1), <math>\alpha</math> at the identity element of <math>G</math> is <math>|G|</math>. By Step (4), it is <math>\sum_{i=1}^r d_i^2</math>. Combining, we get the result.</toggledisplay>
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===Proof in other characteristics===
This follows from the characteristic zero proof, and the fact that [[degrees of irreducible representations are the same for all splitting fields]].
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