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There is a well-known result of Solomon which states that sum of entries of any row in $\mathbb{C}$-character table of a group $G$ is an integer number.

It is mentioned in Martin Isaacs Character Theory of Finite Group as a note that the column sums are also integers. My question is that what's the reason of this latter fact about column sums?

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Let $K$ be the Galois field extension of $\mathbb{Q}$ generated by the entries of the character table. Then the action of $\operatorname{Gal}(K/\mathbb{Q})$ on character values permutes the rows of the character table. So $\operatorname{Gal}(K/\mathbb{Q})$ fixes the sum of each column, which is therefore rational. But the column sums are sums of character values, and therefore algebraic integers, and a rational algebraic integer is just an integer.

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  • $\begingroup$ Thank you for your beautiful answer. I didn't expect the problem to have this kind of approach. So I learned something new :) $\endgroup$ – user97635 Nov 19 '15 at 14:21

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