Minimal polynomial (field theory)

Minimal polynomial (field theory)
In field theory, a branch of mathematics, the minimal polynomial of an element of a field extension is, roughly speaking, the polynomial of lowest degree having coefficients in the field, such that is a root of the polynomial. If the minimal polynomial of 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 and an element of the extension field . The minimal polynomial of an element, if it exists, is a member of , the ring of polynomials in the variable with coefficients in . Given an element of , let be the set of all polynomials in such that . The element is called a root or zero of each polynomial in
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 since for all and . This makes the zero polynomial useless for classifying different values of into types, so it is excepted. If there are any non-zero polynomials in , i.e. if the latter is not the zero ideal, then is called an algebraic element over , and there exists a monic polynomial of least degree in . This is the minimal polynomial of with respect to . It is unique and irreducible over . If the zero polynomial is the only member of , then is called a transcendental element over and has no minimal polynomial with respect to .
Minimal polynomials are useful for constructing and analyzing field extensions. When is algebraic with minimal polynomial , the smallest field that contains both and is isomorphic to the quotient ring , where is the ideal of generated by . 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 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.
Properties
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.
Uniqueness
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.
Irreducibility
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.
To prove this, first observe that any factorization f = gh implies that either g(α) = 0 or h(α) = 0, because f(α) = 0 and F is a field (hence also an integral domain). 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.
Minimal polynomial generates Jα
The minimal polynomial f of α generates the ideal Jα, i.e. every g in Jα can be factorized as g=fh for some h' in F[x].
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). In particular, there is a unique monic generator f, and all generators must be irreducible. When I is chosen to be Jα, for α algebraic over F, then the monic generator f is the minimal polynomial of α.
Examples
Minimal polynomial of a Galois field extension
Given a Galois field extension the minimal polynomial of any not in can be computed asif 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 extensions
Q()
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 − .
Q()
In general, for the quadratic extension given by a square-free , computing the minimal polynomial of an element can be found using Galois theory. Thenin particular, this implies and . This can be used to determine through a series of relations using modular arithmetic.
Biquadratic field extensions
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 unity
The minimal polynomials in Q[x] of roots of unity are the cyclotomic polynomials.
Swinnerton-Dyer polynomials
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.
See also
Ring of integers
Algebraic number field
Minimal polynomials of
References
Pinter, Charles C. A Book of Abstract Algebra. Dover Books on Mathematics Series. Dover Publications, 2010, p. 270–273.
Polynomials
Field (mathematics)