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 J_α 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_α. The set J_α is so named because it is an ideal of F[x]. The zero polynomial, whose every coefficient is 0, is in every J_α 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 J_α, then α is called an algebraic element over F, and there exists a monic polynomial of least degree in J_α. 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_α, 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 sub_α from F[x] to E that substitutes α for x, that is, sub_α(f(x)) = f(α). The kernel of sub_α, ker(sub_α), is the set of all polynomials in F[x] that have α as a root. That is, ker(sub_α) = J_α from above. Since sub_α is a ring homomorphism, ker(sub_α) 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(sub_α) that generates ker(sub_α). Such a polynomial will have least degree among all non-zero polynomials in ker(sub_α), and a(x) is taken to be the unique monic polynomial among these.

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.

Generalization to unique factorization domains

If $R$ is a unique factorization domain, and $\theta$ is algebraic over the field of fractions of $R$, there exists a polynomial annihilating $\theta$ that divides any other polynomial annihilating $\theta$ in $R[X]$. This polynomial, that can be called the minimal polynomial of $\theta$ over $R$, is unique, up to the product by a unit of $R$. Actually, it is the primitive polynomial of minimal degree annihilating $\theta$.

As a simple corollary, if $\varphi$ is a homomorphism $R\to {\bar {R}}$, then $\varphi$ can be extended to a homomorphism $R[\theta ]\to {\bar {R}}[{\bar {\theta }}]$, where ${\bar {\theta }}$ is a solution of the polynomial in ${\bar {R}}[X]$ obtained by applying $\varphi$ on the coefficients of the minimal polynomial of $\theta$ over $R$.

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

• Weisstein, Eric W. "Algebraic Number Minimal Polynomial". MathWorld.
• Minimal polynomial at PlanetMath.
• Pinter, Charles C. A Book of Abstract Algebra. Dover Books on Mathematics Series. Dover Publications, 2010, p. 270-273. ISBN 978-0-486-47417-5