Minimal polynomial (field theory)

For the minimal polynomial of a matrix, see Minimal polynomial (linear algebra).

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 Jα 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 Jα. The set Jα is so named because it is an ideal of F[x]. The zero polynomial, whose every coefficient is 0, is in every Jα since 0αi = 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 Jα, then α is called an algebraic element over F, and there exists a monic polynomial of least degree in Jα. 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 Jα, 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.

Definition

Let E/F be a field extension, α an element of E, and F[x] the ring of polynomials in x over F. The minimal polynomial of α is the monic polynomial of least degree among all polynomials in F[x] having α as a root; it exists when α is algebraic over F, that is, when f(α) = 0 for some non-zero polynomial f(x) in F[x].

Uniqueness

Let a(x) be the minimal polynomial of α with respect to E/F. The uniqueness of a(x) is established by considering the ring homomorphism subα from F[x] to E that substitutes α for x, that is, subα(f(x)) = f(α). The kernel of subα, ker(subα), is the set of all polynomials in F[x] that have α as a root. That is, ker(subα) = Jα from above. Since subα is a ring homomorphism, ker(subα) is an ideal of F[x]. Since F[x] is a principal ring whenever F is a field, there is at least one polynomial in ker(subα) that generates ker(subα). Such a polynomial will have least degree among all non-zero polynomials in ker(subα), and a(x) is taken to be the unique monic polynomial among these.

Alternative proof of uniqueness

Suppose p and q are monic polynomials in Jα of minimal degree n > 0. Since p - q ∈ Jα and deg(p - q) < n it follows that p - q = 0, i.e. p = q.

Properties

A minimal polynomial is irreducible. Let E/F be a field extension over F as above, α ∈ E, and f ∈ F[x] a minimal polynomial for α. Suppose f = gh, where g,h ∈ F[x] are of lower degree than f. Now f(α) = 0. Since fields are also integral domains, we have g(α) = 0 or h(α) = 0. This contradicts the minimality of the degree of f. Thus minimal polynomials are irreducible.

Examples

If F = Q, E = R, α = √2, then the minimal polynomial for α is a(x) = x2 − 2. The base field F is important as it determines the possibilities for the coefficients of a(x). For instance, if we take F = R, then the minimal polynomial for α = √2 is a(x) = x − √2.

If α = √2 + √3, then the minimal polynomial in Q[x] is a(x) = x4 − 10x2 + 1 = (x − √2 − √3)(x + √2 − √3)(x − √2 + √3)(x + √2 + √3).

The minimal polynomial in Q[x] of the sum of the square roots of the first n prime numbers is constructed analogously, and is called a Swinnerton-Dyer polynomial.

The minimal polynomials in Q[x] of roots of unity are the cyclotomic polynomials.

References

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