Complete Factorizations of Proper Element in Principal Ideal Domain are Equivalent
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This theorem requires a proof.
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.
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
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.
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Sources
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 62$. Factorization in an integral domain