Proof of the Fundamental Theorem of Algebra's uniqueness part without utilizing Bézout's lemma.
Let be prime. From or .
Let and a minimal counterexample. We assume, that and show, that .
Theorem (Fundamental Theorem of Arithmetic)
Every has an unique prime factorization.
and a the smallest number with a non-unique decomposition by primes. From Euclid's lemma follows, that for some (s,t) .