Problem 1527 (difficulty: 5/10)

Let \(\displaystyle m>1\) and \(\displaystyle a, b:\Z_m\to\CC\) be two functions. Define the sum \(\displaystyle a+b\) and the convolution \(\displaystyle a*b\) of \(\displaystyle a\) and \(\displaystyle b\) as follows

\(\displaystyle (a+b)(n) = a(n)+b(n); \qquad (a* b)(n) = \sum_{k=0}^{m-1}a(k)b(n-k).\)

Prove that this makes the set of complex valued functions on \(\displaystyle \Z_m\) a commutative ring with unit.

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