# Complete Factorizations of Proper Element in Principal Ideal Domain are Equivalent

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## 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