minimal polynomial

Let a be an element of some field F and let k be a subfield of F. Suppose that a is a root of some nonzero polynomial in k[x]. The minimal polynomial of a over k is the monic (i.e. xⁿ+lower degree terms) polynomial of least degree in k[x] that has a as a root.

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.

Some more funky facts about minimal polynomials:

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].