Polynomial Ring R[x] Construction

Abstract Algebra · Axiom Academy

The formal construction of polynomials over a ring: from sequences with finite support to a rich algebraic structure with operations, degrees, and special properties. This might seem strange at first—a polynomial is a sequence ? But this formalization avoids circular reasoning about "x" and makes operations precise. 2. Degree and Leading Coefficient Two fundamental properties characterize each polynomial: f(x) = 3x² + 2x + 1 has deg(f) = 2 and LC(f) = 3 g(x) = 7 (constant) has deg(g) = 0 and LC(g) = 7 h(x) = x⁵ - 3x³ + x has deg(h) = 5 and LC(h) = 1 Addition in R[x] is defined coefficient-wise: Multiplication is more intricate—it uses the distributive law and combines terms: The properties of R[x] closely mirror those of R:

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