Let be an algebraic integer in an algebraic number field . Let denote the norm over . Prove that the number of factors in any irreducible factorization of over is at most .
Note that is some real number between and , where . No natural number less than can have prime factors (including multiplicity), and since is multiplicative, the number of irreducible factors of is bounded above by the number of prime factors of . So the number of irreducible factors of in is at most .