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## Homework Statement

Let q be the number of units in finite ring R. Show that for all a in R, if a is a unit in R then [tex]a^q = 1[/tex].

Is there a way to solve this without using group theory? All I can seem to find information on is when a and m are relatively prime then [tex]a^{\phi (m)} = 1 (mod \, m)[/tex], which I'd like to prove using the problem I can't solve.

## Homework Equations

## The Attempt at a Solution

I really haven't been able to get anywhere on this. Are there certain patterns that finite rings always follow, that I can exploit?

Thanks