In a polynomial ring over a field, greatest common divisors exist and are unique up to units and Bezout’s identity holds

Let F be a field and let a(x),b(x) \in F[x]. Then a greatest common divisor d(x) of a(x) and b(x) exists and is unique up to multiplication by a unit. Moreover, there exist s(x) and t(x) such that d = as+bt.


This follows because F[x] is a Euclidean domain.

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