Let F be a field, let n be a positive integer, and let c(x) E F[x] be a polynomial of degree n. Let m(x) E F[x] be a polynomial satisfying the following two conditions: (1) m(x) divides c(x), (2) if p(x) E F[x] is an irreducible factor of c(x), then p(x) divides m(x). Decide whether the following statement is true or false: there exists an n × n matrix over F that has characteristic polynomial c(x) and minimal polynomial m(x). If you think it is true, give a proof, and if you think it is false, give a counterexample.

Let F be a field, let n be a positive integer, and let c(x) E F[x] be a polynomial of degree n. Let m(x) E F[x] be a polynomial satisfying the following two conditions: (1) m(x) divides c(x), (2) if p(x) E F[x] is an irreducible factor of c(x), then p(x) divides m(x). Decide whether the following statement is true or false: there exists an n × n matrix over F that has characteristic polynomial c(x) and minimal polynomial m(x). If you think it is true, give a proof, and if you think it is false, give a counterexample.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Let F be a field, let n be a positive integer, and let c(x) E F[x] be a polynomial of degree
n. Let m(x) E F[x] be a polynomial satisfying the following two conditions:
(1) m(x) divides c(x),
(2) if p(x) E F[x] is an irreducible factor of c(x), then p(x) divides m(x).
Decide whether the following statement is true or false: there exists an n x n matrix over F
that has characteristic polynomial c(x) and minimal polynomial m(x). If you think it is true,
give a proof, and if you think it is false, give a counterexample.

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