What is "minimal polynomial"

n. 1 (context linear algebra English) Given a square matrix ''M'' over a field ''K'', the ''minimal polynomial'' is the monic polynomial over ''K'' of smallest degree, which when applied to ''M'' yields the zero matrix. 2 (context field theory English) Given a number ''α'' which belongs to some extension field of a field ''K'' and is algebraic over ''K'', the ''minimal polynomial'' for ''α'' over ''K'' is the monic polynomial over ''K'' of smallest degree for which ''α'' is a root.

Minimal polynomial may refer to either of two closely related concepts:

• Minimal polynomial (linear algebra), the monic polynomial p(x) of least degree such that p(A) = 0 for a square matrix A
• Minimal polynomial (field theory), the monic polynomial p(x) over a field F of least degree such that p(α) = 0, for an algebraic element α over a field F

In field theory, a branch of mathematics, a minimal polynomial is defined relative to a field extension E/F and an element of the extension field E. The minimal polynomial of an element, if it exists, is a member of F[x], the ring of polynomials in the variable x with coefficients in F. Given an element α of E, let J be the set of all polynomials f(x) in F[x] such that f(α) = 0. The element α is called a root or zero of each polynomial in J. The set J is so named because it is an ideal of F[x]. The zero polynomial, whose every coefficient is 0, is in every J since 0α = 0 for all α and i. This makes the zero polynomial useless for classifying different values of α into types, so it is excepted. If there are any non-zero polynomials in J, then α is called an algebraic element over F, and there exists a monic polynomial of least degree in J. This is the minimal polynomial of α with respect to E/F. It is unique and irreducible over F. If the zero polynomial is the only member of J, then α is called a transcendental element over F and has no minimal polynomial with respect to E/F.

Minimal polynomials are useful for constructing and analyzing field extensions. When α is algebraic with minimal polynomial a(x), the smallest field that contains both F and α is isomorphic to the quotient ring F[x]/⟨a(x)⟩, where ⟨a(x)⟩ is the ideal of F[x] generated by a(x). Minimal polynomials are also used to define conjugate elements.

In linear algebra, the minimal polynomial of an matrix over a field is the monic polynomial over of least degree such that . Any other polynomial with is a (polynomial) multiple of .

The following three statements are equivalent:

1. is a root of ,

2. is a root of the characteristic polynomial of ,

3. is an eigenvalue of matrix .

The multiplicity of a root of is the largest power such that strictly contains . In other words, increasing the exponent up to will give ever larger kernels, but further increasing the exponent beyond will just give the same kernel.

If the field is not algebraically closed, then the minimal and characteristic polynomials need not factor according to their roots (in ) alone, in other words they may have irreducible polynomial factors of degree greater than . For irreducible polynomials one has similar equivalences:

1. divides ,

2. divides ,

3. the kernel of has dimension at least .
4. the kernel of has dimension at least .

Like the characteristic polynomial, the minimal polynomial does not depend on the base field, in other words considering the matrix as one with coefficients in a larger field does not change the minimal polynomial. The reason is somewhat different from for the characteristic polynomial (where it is immediate from the definition of determinants), namely the fact that the minimal polynomial is determined by the relations of linear dependence between the powers of : extending the base field will not introduce any new such relations (nor of course will it remove existing ones).

The minimal polynomial is often the same as the characteristic polynomial, but not always. For example, if is a multiple of the identity matrix, then its minimal polynomial is since the kernel of is already the entire space; on the other hand its characteristic polynomial is (the only eigenvalue is , and the degree of the characteristic polynomial is always equal to the dimension of the space). The minimal polynomial always divides the characteristic polynomial, which is one way of formulating the Cayley–Hamilton theorem (for the case of matrices over a field).