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.

Clearly the set of continuous function \( C \left [ a, b \right ] \) is a vector space. Now, we have to prove that the function \( \left | \left | \cdot \right | \right | _{\infty} :  C \left [ a, b \right ] \to \mathbb{R}  \) such that  \( \left | \left | f \right | \right | _{\infty} = \displaystyle \max_{a\leq x\leq b} \left| f(x) \right| \) for every function \( f \in C \left [ a, b \right ] \) is really a norm.

 Generate a random species.