Polynomials are fundamental objects in algebra, but unfortunately most accounts of them skimp on giving a proper definition. Here we base polynomials on the more basic objects of polynumbers. This lecture is part of the MathFoundations series, which tries to lay out proper foundations for mathematics, and will not shy away from discussing the serious logical difficulties entwined in modern pure mathematics. We introduce the particular positive polynumber alpha, and show that any polynumber can be written as a linear combination of powers of alpha. Then we define a positive polynomial to be a positive polynumber written in this standard alpha form.
Does mathematics make logical sense? No, it does not. Foundational issues have been finessed by modern mathematicians, and this series aims to turn things around. And it will have interesting things to say also about mathematics education---especially at the primary and high school level. The plan is to start right from the beginning, and to define all the really important concepts of basic mathematics without any waffling or appeals to authority. Roughly we discuss first arithmetic, then geometry, then algebra, then analysis, then set theory. This course is aimed for a general audience, interested in mathematics, or willing to learn.