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.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter8: Polynomials

Section8.3: Factorization In F [x]

Problem 2E: Let Q denote the field of rational numbers, R the field of real numbers, and C the field of complex....

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