Minimal polynomial (field theory)
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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 solution of the equation in which the polynomial is equated to zero. If the minimal polynomial of α exists, it is unique. The coefficient of the highestdegree 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 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, all of whose coefficients are 0, is in every J_{α} since 0α^{i} = 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 nonzero 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.
Contents
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 nonzero 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 nonzero polynomials in ker(sub_{α}), 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 J_{α} of minimal degree n > 0. Since p − q ∈ J_{α} 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) = x^{2} − 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) = x^{4} − 10x^{2} + 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 SwinnertonDyer 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.org.
 Pinter, Charles C. A Book of Abstract Algebra. Dover Books on Mathematics Series. Dover Publications, 2010, p. 270–273. ISBN 9780486474175