The Missing Spectral Basis in Algebra and Number Theory
Garret Sobczyk · American Mathematical Monthly · 2001
For out of olde feldes, as men seith, Cometh al this newe corne fro yeere to yere; And out of olde bokes, in good feith, Cometh al this new science that men lere. –Chaucer 1. BEGINNINGS In Euclid’s Elements, Book VII we find Proposition 2: Given two numbers not prime to one another, to find their greatest common measure. Then follows what mathematicians refer to as the Euclidean algorithm [4, p. 298]. We need the following consequence of this venerable Proposition. Given r positive inte-gers h1, h2,..., hr ∈ N whose greatest common divisor is 1 ∈ N, there exist integers b1, b2,..., br ∈ Z with the property that b1h1 + b2h2 + · · · + br hr = 1. (1) The justified fame of the Euclidean algorithm derives from the fact that it has a much larger realm of applicability than just the integers. In particular, let K be any field and letK[x] be the corresponding integral domain of polynomials overK. Given r polynomials h1(x), h2(x),..., hr (x) ∈ K[x] whose greatest common divisor is 1 ∈ K (no common zeros inK), there exist polynomials b1(x), b2(x),..., br(x) ∈ K[x] with the property that b1(x)h1(x)+ b2(x)h2(x)+ · · · + br (x)hr(x) = 1. (2) The identities (1) and (2), and the striking analogy between them, provide the grist for this article. 2. MODULAR NUMBERS Let Zh = {0, 1, 2,..., h − 1} be the modular number system modulo h, where h ∈ N. Of course, the numbers b ∈ Zh represent equivalence classes modulo h and addition, multiplication, and equality in Zh are defined modulo h. Thus, when we write b + c = d and bc = d in Zh, we mean that b + c ≡ d mod(h) and bc ≡ d mod(h). The modular number system Zh is isomorphic to the factor ring Z/ for the principal ideal [2, p. 248]. By unique factorization, we can write h = pm11 pm22 · · · pmrr where each pi is a distinct prime factor of h, and we can order the factors pmii so that their multiplicities satisfy 1 ≤ m1 ≤ m2 ≤ · · · ≤ mr. For a given h ∈ N, define hi: = h/pmii for i = 1,..., r. Since the hi have no com-mon factor other than 1, invoking (1) gives b1h1 + b2h2 + · · · + br hr = 1. (3) Whereas this equation holds in Z, it is just as valid when interpreted as an identity in Zh. Defining the numbers si: = bi hi ∈ Zh, we say that sb(Zh): = {s1, s2,..., sr} is the