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Minimal polynomial

The minimal polynomial of an n-by-n matrix A over a field F is the monic polynomial p(x) over F of least degree such that p(A)=0.

The following three statements are equivalent:

  1. λ∈F is a root of p(x),
  2. λ is a root of the characteristic polynomial of A,
  3. λ is an eigenvalue of A.

The multiplicity of a root λ of p(x) is the geometrical multiplicity of λ and is the size of the largest Jordan block corresponding to λ.

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In field theory, a minimal polynomial is a polynomial m(x) in the field Zp (with p prime), such that, if we have the field F=Zp(α), it is the polynomial of least degree with m(α)=0.

The minimal polynomial is unique, and if we have some irreducible polynomial f(x) with f(α)=0, then f is the minimal polynomial of α.