Minimal polynomial (field theory)

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.

Definition

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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

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Throughout this section, let E/F be a field extension over F as above, let αE be an algebraic element over F and let Jα be the ideal of polynomials vanishing on α.

Uniqueness

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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 := fgJα (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 cmF for the non-zero coefficient of highest degree in r) is a monic polynomial of degree m < n such that r / cmJα (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 = fg, i.e. that f = g.

Irreducibility

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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

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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

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Minimal polynomial of a Galois field extension

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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 extensions

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Q(2)

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If F = Q, E = R, α = 2, 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 = 2 is a(x) = x2.

Q(d )

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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 extensions

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If α = 2 + 3, then the minimal polynomial in Q[x] is a(x) = x4 − 10x2 + 1 = (x23)(x + 23)(x2 + 3)(x + 2 + 3).

Notice if   then the Galois action on   stabilizes  . Hence the minimal polynomial can be found using the quotient group  .

Roots of unity

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The minimal polynomials in Q[x] of roots of unity are the cyclotomic polynomials. The roots of the minimal polynomial of 2cos(2pi/n) are twice the real part of the primitive roots of unity.

Swinnerton-Dyer polynomials

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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

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References

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  • Weisstein, Eric W. "Algebraic Number Minimal Polynomial". MathWorld.
  • Minimal polynomial at PlanetMath.
  • Pinter, Charles C. A Book of Abstract Algebra. Dover Books on Mathematics Series. Dover Publications, 2010, p. 270–273. ISBN 978-0-486-47417-5