minimal polynomial

Here are some properties of the minimal polynomial

• It is unique. (For suppose that f and h are both minimal polynomials of a over k. Then f-h has lower degree than f and h and has a as a root. If it is not zero this contradicts the definition of minimal polynomial.)
• It is irreducible. (If not then one of its factors has smaller degree and has a as a root, again contradicting the definition.)
• If h(x) is a nonzero polynomial over k that has a as a root then the minimal polynomial is a factor of h(x). (Similarly.)

  Examples

  • The minimal polynomial of 2^1/2 over Q (the rational numbers) is x²-2. More generally, 2^1/n has minimal polynomial xⁿ-2 over Q. (See Eisenstein's irreducibility criterion.)
  • A primitive nth root of unity has the nth cyclotomic polynomial as its minimal polynomial over Q.

  The Cayley-Hamilton Theorem shows that an nxn matrix is a zero of a polynomial. It follows that there is an analogous notion of minimal polynomial for matrices. The derivation of the Jordan canonical form for matrices uses the minimal polynomial.

Let F ⊆ E be fields, and α ∈ E. We denote F[α] to be the smallest subring of E containing both F and α, and F(α) to be the smallest subfield containing both F and α. Let n(x) ∈ F[x] be the minimal polynomial of α over F.

Obviously F(α) ⊆ F[α], since all subfields are subrings. Now F(α) = F[α] iff α is algebraic over F (i.e. n(x) exists).

Also, F(α) ≅ F[x] / n(x)F[x].