Associates in Ring of Polynomial Forms over Field
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Theorem
Let $F \left[{X}\right]$ be the ring of polynomial forms over the field $F$.
Then $d \left({X}\right) \in F \left[{X}\right]$ is an associate of $d' \left({X}\right)$ iff $d \left({X}\right) = c \cdot d' \left({X}\right)$ for some $c \in F, c \ne 0$.
Hence any two polynomials in $F \left[{X}\right]$ have a unique monic GCD.
Proof
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Sources
- C.R.J. Clapham: Introduction to Abstract Algebra (1969)... (previous)... (next): $\S 6.28$: Example $55$
