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Algebraic integers

Among the properties of characters which may be regarded as fundamental, perhaps the most opaque is that which states that the degree of an irreducible character of a finite group *G* must divide the order of *G*. This is one of several results which we shall prove in this chapter, using algebraic integers. Most of the results concern arithmetic properties of character values. We discuss properties of a group element *g* ∈ *G* which ensure that *χ*(*g*) is an integer for all characters *χ* of *G*. And we prove some useful congruence properties; for example, if *p* is a prime number and *g* ∈ *G* is an element of order *p ^{r}* for some

**Algebraic integers**

*22.1 Definition ...*

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