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 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 1

Properties 2

Examples 3

References 4
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.
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. 270273. ISBN 9780486474175
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