The theorem If and are such that is separately coprime with each , then is coprime with their product . The proof Proof: Through Bézout. being coprime with each , we can write, for each : If we take the product of each of these expressions, we get: Among the terms of this product, all…
For the definition and characterization of the polynomials over a field , see Polynomials over a field. Definition of the degree of a polynomial The degree of a polynomial P, written , is an element of . The symbol is taken to have the usual properties of order and of addition relative to natural numbers….
In this context, we consider a commutative field (simply: field) . The polynomials will be constructed as a certain associative unital algebra over (“-aua”), together with a distinguished element called the formal variable. itself, as a vector space over itself and the usual multiplication in as the third law, is a -aua, Morphisms between two…
Definition Let be a commutative field. The polynomials form a commutative unital algebra over containing an element such that the following universal property holds: For any unital algebra (commutative or otherwise) and any element , there exists exactly one unital algebra morphism such that . If and are elements of , let be the unique…