Euclid's Lemma for Euclidean Domains
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Theorem
Let $\struct {D, +, \times}$ be a Euclidean domain whose unity is $1$.
Let $a, b, c \in D$.
Let $a \divides b \times c$, where $\divides$ denotes divisibility.
Let $a \perp b$, where $\perp$ denotes relative primeness.
Then $a \divides c$.
Proof
| \(\ds a\) | \(\perp\) | \(\ds b\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \gcd \set {a, b}\) | \(=\) | \(\ds 1\) | Definition of Coprime Elements of Euclidean Domain | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \exists x, y \in D: \, \) | \(\ds a \times x + b \times y\) | \(=\) | \(\ds 1\) | Bézout's Identity on Euclidean Domain | |||||||||
| \(\ds \leadsto \ \ \) | \(\ds c\) | \(=\) | \(\ds c \times \paren {a \times x + b \times y}\) | |||||||||||
| \(\ds \) | \(=\) | \(\ds c \times a \times x + c \times b \times y\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {a \times c} \times x + \paren {b \times c} \times y\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds a\) | \(\divides\) | \(\ds c \times \paren {a \times x + b \times y}\) | as $a \divides a \times c$ and $a \divides b \times c$ | ||||||||||
| \(\ds \) | \(=\) | \(\ds c \times 1\) | Bézout's Identity on Euclidean Domain | |||||||||||
| \(\ds \) | \(=\) | \(\ds c\) |
$\blacksquare$
Source of Name
This entry was named for Euclid.
Also see
- Euclid's Lemma, for the usual statement of this result, which is this lemma as applied specifically to the integers.
Sources
- 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $6$: Polynomials and Euclidean Rings: $\S 29$. Irreducible elements: Theorem $55$