Question

Let set S denote the set of all polynomials in one variable with real coefficients. For example, 3+21​x+7x2+4x3∈S S has a familiar definition of addition and multiplication. Moreover, S has an additive identity, the constant polynomial 0 , and a multiplicative identity, the constant polynomial 1. Is S a field? Explain your answer

Answer 1

The set S of all polynomials with real coefficients is not a field because higher-degree polynomials do not have multiplicative inverses within S.

Step-by-step explanation:

The set S, representing all polynomials with real coefficients, does have an additive identity (the constant polynomial 0) and a multiplicative identity (the constant polynomial 1). However, for S to be a field, every non-zero element must have a multiplicative inverse within S. While the additive properties, such as commutativity (A + B = B + A), and the presence of the identity elements align with the properties of a field, the set S fails to satisfy the multiplicative inverse property for higher-degree polynomials. For example, x^2 has no polynomial within S that can serve as its inverse (such that x^2 multiplied by this inverse yields 1). Thus, S is not a field.