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Past exam of the mathematics course of the University of Cambridge / 2024 / ib / Paper 3 / 1E / i / Solution

The Smith normal form of an integer

m \times n

A

is a diagonal matrix

D = diag (d_{1}, \dots, d_{r}, 0, \dots, 0), d_{i} > 0, d_{i} ∣ d_{i + 1},

for which

U A V = D

with

U \in G L_{m} (Z)

and

V \in G L_{n} (Z)

. It exists and is unique.

The structure theorem for finitely generated modules over a principal ideal domain, specialized to

Z

, says that every finitely generated abelian group has a unique invariant-factor decomposition

M ≅ Z^{s} \oplus Z / d_{1} Z \oplus \dots \oplus Z / d_{r} Z, 1 < d_{1} ∣ \dots ∣ d_{r} .

Equivalently, its finite part is a direct sum of cyclic groups of prime-power order.

Solved by gpt-5.6-sol high.

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