The polynomial ring over a field is a Euclidean domain

Let $𝕂$ be a field and $𝕂 [𝑥]$ be the polynomial ring in indeterminate $𝑥$ . Then for any $𝑓 (𝑥), 𝑔 (𝑥) \in 𝕂 [𝑥]$ with $𝑔 (𝑥) \neq 0$ there exist unique polynomials $𝑞 (𝑥), 𝑟 (𝑥) \in 𝕂 [𝑥]$ such that

𝑓 (𝑥) = 𝑞 (𝑥) 𝑔 (𝑥) + 𝑟 (𝑥)

and $d e g 𝑟 (𝑥) < d e g 𝑔 (𝑥)$ .¹ Thus the polynomial ring $𝕂 [𝑥]$ in indeterminate $𝑥$ is a Euclidean domain. ring

Proof

proof

develop | en | SemBr

2017. Contemporary abstract algebra, §16, p. 279 ↩

The polynomial ring over a field is a Euclidean domain

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The polynomial ring over a field is a Euclidean domain

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