176 III More on Divisibility
For proofs of these results and for later developments, see Lam [29], Fitchas and
Galligo [14], and Swan [50]. There is a short proof of the Quillen–Suslin theorem in
Lang [31].
For Smith’s normal form, see Smith [49] and Kaplansky [27]. It was shown by
Wedderburn (1915) that Smith’s normal form also holds for matrices of holomor-
phic functions, even though the latter do not form a principal ideal domain; see
Narasimhan [37].
Finitely generated commutative groups are important, not only because more can
be said about them, but also because they arise in practice.Dirichlet’s unit theorem
says that the units of an algebraic number field form a finitely generated commutative
group. As will be seen in Chapter XIII,§4,Mordell’s theoremsays that the rational
points of an elliptic curve also form a finitely generated commutative group.
Modules over a polynomial ringK[s] play an important role in what electrical
engineers calllinear systems theory. Connected accounts are given in Kalman [26],
Rosenbrock [44] and Kailath [25]. For some further mathematical developments, see
Forney [15], Coppel [11], and Coppel and Cullen [12].
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