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 Incoming links: Linear polynomial

Past exam of the linear algebra course of the University of Oxford / 2023 / 1 / a / i / Solution 2026-10-25

The primary decomposition theorem for linear maps starts with a linear map

T : V \to V

on a finite-dimensional vector space over a field

F

. Factor its minimal polynomial as

m_{T} (x) = \prod_{j = 1}^{r} p_{j} (x)^{a_{j}},

where the

p_{j}

are distinct monic polynomials that are irreducible polynomials and

a_{j} \geq 1

V_{j} = ker p_{j} (T)^{a_{j}}

. Then

V = V_{1} \oplus \dots \oplus V_{r}, T (V_{j}) \subseteq V_{j}, m_{T ∣_{V_{j}}} = p_{j}^{a_{j}} .

Thus the direct sum separates the irreducible polynomial factors into uniquely determined invariant subspaces. The statement works even when the minimal polynomial does not split into linear polynomials over

F

.

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Wheel factorization 2026-10-20

Choose a modulus

W

that is a product of small primes. Every prime not dividing

W

must be coprime to

W

, so a search can restrict candidates to the corresponding residue classes modulo

W

. For several linear polynomials, discard each residue class in which any required value is divisible by a selected prime. The surviving residue classes can be combined using the Chinese remainder theorem.

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