In field theory, a branch of mathematics, the minimal polynomial of an element α of an extension field of a field is, roughly speaking, the polynomial of lowest degree having coefficients in the smaller field, 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.
More formally, a minimal polynomial is defined relative to a field extension E/F and an element of the extension field E/F.
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 More specifically, Jα is the kernel of the ring homomorphism from F[x] to E which sends polynomials g to their value g(α) at the element α.
Because it is the kernel of a ring homomorphism, Jα is an ideal of the polynomial ring F[x]: it is closed under polynomial addition and subtraction (hence containing the zero polynomial), as well as under multiplication by elements of F (which is scalar multiplication if F[x] is regarded as a vector space over F).
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, i.e. if the latter is not the zero ideal, 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 f(x), the smallest field that contains both F and α is isomorphic to the quotient ring F[x]/⟨f(x)⟩, where ⟨f(x)⟩ is the ideal of F[x] generated by f(x).
Minimal polynomials are also used to define conjugate elements.
Throughout this section, let E/F be a field extension over F as above, let α ∈ E be an algebraic element over F and let Jα be the ideal of polynomials vanishing on α.
The minimal polynomial f of α is unique.
To prove this, suppose that f and g are monic polynomials in Jα of minimal degree n > 0.
We have that r := f−g ∈ Jα (because the latter is closed under addition/subtraction) and that m := deg(r) < n (because the polynomials are monic of the same degree).
If r is not zero, then r / cm (writing cm ∈ F for the non-zero coefficient of highest degree in r) is a monic polynomial of degree m < n such that r / cm ∈ Jα (because the latter is closed under multiplication/division by non-zero elements of F), which contradicts our original assumption of minimality for n. We conclude that 0 = r = f − g, i.e. that f = g. The minimal polynomial f of α is irreducible, i.e. it cannot be factorized as f = gh for two polynomials g and h of strictly lower degree.
Choosing both g and h to be of degree strictly lower than f would then contradict the minimality requirement on f, so f must be irreducible.
To prove this, it suffices to observe that F[x] is a principal ideal domain, because F is a field: this means that every ideal I in F[x], Jα amongst them, is generated by a single element f. With the exception of the zero ideal I = {0}, the generator f must be non-zero and it must be the unique polynomial of minimal degree, up to a factor in F (because the degree of fg is strictly larger than that of f whenever g is of degree greater than zero).
Note that the same kind of formula can be found by replacing
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.
In general, for the quadratic extension given by a square-free
, computing the minimal polynomial of an element
through a series of relations using modular arithmetic.
Hence the minimal polynomial can be found using the quotient group
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.