Prime factorization is unique

Prime factorization is unique is, in the context of the integers, part of the fundamental theorem of arithmetic. But the claim can be proven more generally in the context of abstract algebra.

The uniqueness here refers to uniqueness except for multiplication with a unit. In the case of the integers the units are [math]\displaystyle{ 1 }[/math] and [math]\displaystyle{ -1 }[/math], so [math]\displaystyle{ 6=2\cdot 3 = (-2)\cdot (-3) }[/math] is considered a unique factorization even though [math]\displaystyle{ -2, -3 }[/math] also have the properties of prime elements. For integers this relative uniqueness is turned into real uniqueness by requiring prime numbers to be positive.

Proof

Integers

Let [math]\displaystyle{ a }[/math] be an integer and [math]\displaystyle{ a= p_1\cdot\text{ . . . }\cdot p_r = \left(-1\right)^l q_1\cdot\text{ . . . }\cdot q_s }[/math] two prime factorizations of [math]\displaystyle{ a }[/math]. The prime factor [math]\displaystyle{ p_1 }[/math] on the left divides the product [math]\displaystyle{ \left(-1\right)^l q_1\cdot\text{ . . . }\cdot q_s }[/math] on the right, so according to Euclid's lemma (If a prime divides a product it divides one of the factors) [math]\displaystyle{ p_1 }[/math] divides one of the factors [math]\displaystyle{ q_i }[/math] (no prime divides [math]\displaystyle{ \pm 1 }[/math]). So [math]\displaystyle{ p_1 = q_i }[/math] or [math]\displaystyle{ p_1 = -q_i }[/math]. If we change the order of the [math]\displaystyle{ q_1\text{, . . . , }q_s }[/math] we can assume [math]\displaystyle{ p_1 = q_1 }[/math] or [math]\displaystyle{ p_1 = -q_1 }[/math] and after "dividing" by [math]\displaystyle{ p_1 }[/math] on both sides we get [math]\displaystyle{ p_2\cdot\text{ . . . }\cdot p_r = \left(-1\right)^{l'} q_2\cdot\text{ . . . }\cdot q_s }[/math].

By repeating this procedure we can see that [math]\displaystyle{ r=s }[/math] and that, after reordering, [math]\displaystyle{ p_1=\pm q_1\text{, . . . , }p_r=\pm q_r }[/math].

Integral domains

The same proof as above also works generaly for an integral domain with a few tweaks. Note that in the general case prime elements are defined by the property If a prime divides a product it divides one of the factors and not the one used for the integers.

Claim

Statement of the claim	Prime factorization is unique
Level of certainty	Proven
Nature	Theoretical
Counterclaim
Dependent on	If a prime divides a product it divides one of the factors
Dependency of