Complete Factorizations of Proper Element in Principal Ideal Domain are Equivalent

Theorem

Let $\struct {D, +, \circ}$ be a principal ideal domain.

Let $x \in D$ be a proper element of $D$.

Let there be two complete factorizations of $x$:

$x = u_y \circ y_1 \circ y_2 \circ \cdots \circ y_m = F_1$

$x = u_z \circ z_1 \circ z_2 \circ \cdots \circ z_n = F_2$

Then $F_1$ and $F_2$ are equivalent.

Proof

This theorem requires a proof. In particular: Whitelaw leaves this unresolved at the end of $\S 62$ as an exercise for the student. I haven't read ahead that far, but it may be proved in the exercises. Will return to this later. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

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Sources

• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 62$. Factorization in an integral domain