Fraktal

  It is easy to see that \( \mathrm{arg} \left ( \frac{a}{b} \right ) = \mathrm{arg} \left ( a \right )  - \mathrm{arg} \left ( b \right )  \) for any complex number \( a \) and \( b \). But, this is not the case for the principle value that is \( \mathrm{Arg} \) does not always equal to \( \mathrm{Arg} \left ( a \right ) - \mathrm{Arg} \left ( b \right )  \).

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  Let \(n\) be a natural number greater than \(1\). Suppose \( \left ( \mathbb{Z}_n, +,  \cdot \right ) \) be a ring of integers mod \(n\). Then, every element \(a \in \mathbb{Z}_n \) has multiplicative inverse if and only if \(n\) is a prime number. In other words,  \( \left ( \mathbb{Z}_n, +,  \cdot \right ) \) is a field if and only if \(n\) is a prime number.