Unique Factorization Theorem

Theorem
Let $\struct {D, +, \times}$ be a Euclidean domain.

Then $\struct {D, +, \times}$ is a unique factorization domain.

Proof
By the definition of unique factorization domain, we need to show that:

For all $x \in D$ such that $x$ is non-zero and not a unit of $D$:


 * $(1): \quad x$ has a complete factorization in $D$
 * $(2): \quad$ Any two complete factorizations of $x$ in $D$ are equivalent.

Proof of Existence
Proof by Complete Induction:

Let $\nu$ be the Euclidean valuation function on $D$.

For all $n \in \Z_{>0}$, let $\map P n$ be the proposition:
 * every $x \in D$ such that $\map \nu x = n$ is either a unit of $D$ or has a complete factorization in $D$.

That is, it can be written as the product of a finite number of irreducible elements.

Basis for the Induction
If $\map \nu x = \map \nu 1$ then $x$ is a unit of $D$.

So $\map P n$ is true for $n = \map \nu 1$.

This is our basis for the induction.

Induction Hypothesis
Now we need to show that, if $\map P k$ is true for all values of $k < \map \nu x$, then it logically follows that $\map P n$ is true.

So this is our induction hypothesis:
 * Every $x \in D$ such that $\map \nu x < n$ is either a unit of $D$ or can be written as the product of a finite number of irreducible elements.

Then we need to show:
 * Every $x \in D$ such that $\map \nu x = n$ is either a unit of $D$ or can be written as the product of a finite number of irreducible elements.

Induction Step
This is our induction step:

Suppose $x \in D$ such that $\map \nu x = n > \map \nu 1$.

If $x$ is irreducible, then it is the product of a finite number (that is, $1$) of irreducible elements.

If not, then $x = y z$, where neither $b$ nor $c$ are either unit of $D$ or irreducible.

By Euclidean Valuation of Non-Unit is less than that of Product we have that:
 * $\map \nu y < \map \nu x$ and $\map \nu z < \map \nu x$

By the induction hypothesis, we assume the truth of $\map P n$ for all values of $n < \map \nu x$.

Hence we know that both $y$ and $z$ can be written as the product of a finite number of irreducible elements.

Thus we may deduce the same about $x$.

So $\map P n$ has been shown to be true by the Second Principle of Mathematical Induction.

Proof of Equivalence
Now we need to show that any two complete factorizations of $x$ in $D$ are equivalent.

Suppose $p_1 p_2 \ldots p_r = q_1 q_2 \ldots q_s$ where all the $p$s and $q$s are irreducible elements of $D$.

Then:
 * $p_1 \divides q_1 q_2 \ldots q_s$

and by Euclid's Lemma for Irreducible Elements:
 * $p_1 \divides q_i$ for some $i$.

Since each of these is irreducible, they must by definition either be associates or that $p_1 = q_i$.

Thus we can write $q_i = u_i p_1$ where $u_i$ is a unit of $D$.

Cancelling $p_1$ from both sides, we continue similarly with $p_2$, and so on.

After a finite number of steps we determine that $r = s$ and that $q_1, q_2, \ldots, q_s$ are just associates of $p_1, p_2, \ldots p_r$ perhaps in a different order.

It follows directly, by definition, that $p_1 p_2 \ldots p_r$ and $q_1 q_2 \ldots q_s$ are equivalent factorizations.

Also see

 * Fundamental Theorem of Arithmetic: the result in the domain of integers