- #1
Mr Davis 97
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I've always been curious about why we define polynomials the way we do. On the surface, it seems that they are expressions that naturally arise from combining the standard arithmetic operations on indeterminates. However, there are some points that I am generally confused about. Why are polynomials defined to have only powers of ##x## that are non-negative integers? Why not rational or negative powers? Also, in studying polynomials, they invariably have coefficients in ##\mathbb{Q}##. Why do we seem to be not as concerned with polynomials with coefficients in ##\mathbb{R}## or ##\mathbb{C}##? For example, in abstract algebra in studying extension fields, the coefficients of polynomials under study are most always assumed to be in ##\mathbb{Q}##, with ##\mathbb{R}## or ##\mathbb{C}## as the extension fields of ##\mathbb{Q}##. Also, the ideas of transcendental and algebraic numbers seem to be defined in terms of polynomials with coefficients in ##\mathbb{Q}##