Minimal polynomial (field theory)
In field theory, a branch of mathematics, the minimal polynomial of a value α is, roughly speaking, the polynomial of lowest degree having coefficients of a specified type, such that α is a root of the polynomial. If the minimal polynomial of α exists, it is unique. The coefficient of the highest-degree term in the polynomial is required to be 1, and the specified type for the remaining coefficients could be integers, rational numbers, real numbers, or others.
More formally, 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, all of whose coefficients are 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.
Let E/F be a field extension, α an element of E, and F[x] the ring of polynomials in x over F. The element α has a minimal polynomial when α is algebraic over F, that is, when f(α) = 0 for some non-zero polynomial f(x) in F[x]. Then the minimal polynomial of α is defined as the monic polynomial of least degree among all polynomials in F[x] having α as a root.
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 uniquenessEdit
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.
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.
Minimal polynomial of a Galois field extensionEdit
Given a Galois field extension the minimal polynomial of any not in can be computed as
if has no stabilizers in the Galois action. Since it is irreducible, which can be deduced by looking at the roots of , it is the minimal polynomial. Note that the same kind of formula can be found by replacing with where is the stabilizer group of . For example, if then its stabilizer is , hence is its minimal polynomial.
Quadratic field extensionsEdit
If F = Q, E = R, α = √, 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 = √ is a(x) = x − √.
In general, for the quadratic extension given by a square-free , computing the minimal polynomial of an element can be found using Galois theory. Then
in particular, this implies and . This can be used to determine through a series of relations using modular arithmetic.
Biquadratic field extensionsEdit
If α = √ + √, then the minimal polynomial in Q[x] is a(x) = x4 − 10x2 + 1 = (x − √ − √)(x + √ − √)(x − √ + √)(x + √ + √).
Notice if then the Galois action on stabilizes . Hence the minimal polynomial can be found using the quotient group .
Roots of unityEdit
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.