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Let $K$ be a commutative field, $E$ a $K$ vector space , and $u$ a linear map over $E$ , suppose that the minimal polynomial of $u$ is irreductible Show that $E$ is a direct sum of cyclic vector spaces

Seginus
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  • Hint: the characteristic polynomial is a power of the minimal polynomial – lhf Mar 31 '17 at 11:25
  • I don't see why is that true – Seginus Mar 31 '17 at 13:37
  • The characteristic polynomial and the minimal polynomial have the same irreducible factors. See http://math.stackexchange.com/questions/825848/showing-that-minimal-polynomial-has-the-same-irreducible-factors-as-characterist and http://math.stackexchange.com/questions/237486/understanding-minimum-polynomials-and-characteristic-polynomials. – lhf Mar 31 '17 at 18:02
  • thank you sir for your answer , but how can you use it to solve the problem ? – Seginus Apr 02 '17 at 19:28
  • more hints please sir ? you use the lemma decomposition des noyaux ? – Seginus Apr 03 '17 at 22:03

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