Fraktal

  Let \( K \) be a finite field. \( 1 + 1 = 0 \) in K if and only if for any \( f \in K \left [ x \right ] \) such that the degree of \( f \) is more or equal to 1, the polynomial \( f(X^2) \) is a reducible polynomial.

  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.

 Decompose any permutation into cycles and transpositions.

Welcome to Permutation Composition Calculator Alpha Version 1.0. This tool can calculate the result of compositions of some permutations in S n bottom-to-top.