A *root of unity* is a complex number such that for some positive integer . This means that for some integer ; if is relatively prime to , we say that is a *primitive* -th root of unity, meaning that is not a -th root of unity for any .

Here’s a question: when can we have

if are roots of unity?

This is a little bit vague, in that I did not say what kind of tentative characterization we are looking for. If you were inclined to play devil’s advocate, you could say that equation (1) provides a good enough description. There are, however, less obvious answers that have come in handy in various parts of my research, so let’s look at some of them.

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