Basics on Polynomials
This chapter presents the fundamental properties of polynomials and, more generally, formal power series. Besides standard topics such as the evaluation of polynomials, roots, formal derivatives, interpolation and the like, we also consider two more advan
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Basics on Polynomials
Abstract This chapter presents the fundamental properties of polynomials and, more generally, formal power series. Besides standard topics such as the evaluation of polynomials, roots, formal derivatives, interpolation and the like, we also consider two more advanced topics which will be important for later chapters, namely M¨obius inversion and endomorphisms of vector spaces and their minimal polynomials.
2.1 Formal Power Series and M¨obius Inversion If one wants to define polynomials in a precise manner, one needs to introduce them as a special type of formal power series. Therefore, we first consider these more general objects in the present section; the basic ingredients are a commutative ring (R, +, ·, 0, 1) and a suitable partially ordered set. We begin by recalling a few wellknown facts concerning the set RX of all mappings from X to R, where X is an arbitrary nonempty set (independent of any ordering). With respect to the pointwise addition defined by ( f + g)(x) := f (x) + g(x)
for all x ∈ X and all f , g ∈ RX ,
(2.1)
RX becomes an abelian group. Here the zero element is the zero mapping with x → 0 for all x ∈ R, and the additive inverse of f ∈ RX is the mapping − f defined by (− f )(x) := − f (x). One also defines a scalar multiplication by r f : X → R, x → r f (x)
for all r ∈ R and f ∈ RX ,
(2.2)
which turns RX into an R-module. Of course, RX can also be made into a commutative ring in a natural way, with respect to the pointwise multiplication given by
© Springer Nature Switzerland AG 2020 D. Hachenberger and D. Jungnickel, Topics in Galois Fields, Algorithms and Computation in Mathematics 29, https://doi.org/10.1007/978-3-030-60806-4_2
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2 Basics on Polynomials
( f · g)(x) := f (x) · g(x)
for all x ∈ X and all f , g ∈ RX .
(2.3)
Moreover, the pointwise multiplication and the scalar multiplication are compatible in the sense that r( f · g) = (r f ) · g = f · (rg) for all r ∈ R and all f , g ∈ RX .
(2.4)
When the underlying ring is actually a field F (so that F X is a vector space), one speaks of an F-algebra. In order to define formal power series, one does not use the pointwise multiplication, but another type of multiplication, the so-called convolution. For this, the set X has to be equipped with a suitable partial order. We change our notation accordingly and also recall some facts from Section 1.2. Definition 2.1.1. Let (N, ·, 1N ) be a (commutative) monoid with cancellation. We say that N is simple, provided that 1N is the only unit in N. Recall from Section 1.2 that the divisibility relation | gives a partial order on any simple monoid N with cancellation. For example, N might arise as the factor monoid M = M/U(M) of some arbitrary commutative monoid (M, ·, 1M ) with cancellation, where U(M) is the group of units of M. Moreover, if (M, |) is locally finite (see Definition 1.2.8), then (N, |) is again locally finite, and, if (M, ·, 1M ) is factorial, then (N, ·, 1N ) is likewise factorial. By Theorem 1.2.14, the property of being factorial implies the local
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