ABSTRACT
22.1 Perhaps ℤ is the most obvious example of a ring, but the next one should be the ring of polynomials over some field. However, the notion of polynomial depends heavily on the notion of variable and a rigorous definition of “variable” may be less trivial than what appears on first sight. On the other hand, we have a good notion of product of groups at hand, so, starting from this, we first define rings of formal power series (22.9) using the notion of “variable” as or indeterminate as defined in 22.7. By truncating formal power series, that is looking at those only involving a finite number of non-zero coefficients, we arrive at a definition of polynomials in one indeterminate (22.11). The advantage of this procedure is that we have done the construction over an arbitrary, even possibly noncommutative, ring. The usual properties of polynomials, including the gradation, are then derived in a straightforward way (cf. 22.13, 22.16 and 22.18). The intuitive notion of indeterminate, or variable, is then in correspondence with the universal properties mentioned in Theorem 22.21. The characterization of when an element in a commutative ring R is a zero of a certain polynomial in R[X] is given in Theorem 22.26 as a consequence of Ruffini's, Bezout - Theorem (22.25) This leads to the important result that a polynomial of degree n over a commutative domain can have at most n zero's in that domain (Theorem 22.28). This chapter is important for the understanding of Galois theory in Chapter 28, Chapter 30.
