Wednesday, 13 July 2016

linear algebra - How to prove the uniqueness of the minimal polynomial?



I have here the definition:




Let T be a linear operator on a finite-dimensional vector
space V over the field F. The minimal polynomial for T is the (unique)

monic generator of the ideal of polynomials over F which annihilate T.




I would like to know how to prove the uniqueness of it, how would I start?


Answer



You can start with proving




If f(T)=g(T)=0 for some polynomials f,g, then gcd





and then the uniqueness will follow from a simple proof by contradiction.


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