Minimal polynomial (field theory)

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 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, and the specified type for the remaining coefficients could be integers, rational numbers, real numbers, or others.

For the minimal polynomial of a matrix, see Minimal polynomial (linear algebra).

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 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 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.

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.

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 non-zero 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

Minimal polynomial of a Galois field extension

Given a Galois field extension ${\displaystyle L/K}$ the minimal polynomial of any ${\displaystyle \alpha \in L}$ not in ${\displaystyle K}$ can be computed as

${\displaystyle f(x)=\prod _{\sigma \in {\text{Gal}}(L/K)}(x-\sigma (\alpha ))}$

if ${\displaystyle \alpha }$ has no stabilizers in the Galois action. Since it is irreducible, which can be deduced by looking at the roots of ${\displaystyle f'}$, it is the minimal polynomial. Note that the same kind of formula can be found by replacing ${\displaystyle G={\text{Gal}}(L/K)}$ with ${\displaystyle G/N}$ where ${\displaystyle N={\text{Stab}}(\alpha )}$ is the stabilizer group of ${\displaystyle \alpha }$. For example, if ${\displaystyle \alpha \in K}$ then its stabilizer is ${\displaystyle G}$, hence ${\displaystyle (x-\alpha )}$ is its minimal polynomial.

Quadratic field extensions

Q(√2)

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) = x − √2.

Q(√d)

In general, for the quadratic extension given by a square-free ${\displaystyle d}$, computing the minimal polynomial of an element ${\displaystyle a+b{\sqrt {d}}}$ can be found using Galois theory. Then

${\displaystyle {\begin{aligned}f(x)&=(x-(a+b{\sqrt {d}}))(x-(a-b{\sqrt {d}}))\\&=x^{2}-2ax+(a^{2}-b^{2}d)\end{aligned}}}$

in particular, this implies ${\displaystyle 2a\in \mathbb {Z} }$ and ${\displaystyle a^{2}-b^{2}d\in \mathbb {Z} }$. This can be used to determine ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {d}})}}$ through a series of relations using modular arithmetic.

Biquadratic field extensions

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

Notice if ${\displaystyle \alpha ={\sqrt {2}}}$ then the Galois action on ${\displaystyle {\sqrt {3}}}$ stabilizes ${\displaystyle \alpha }$. Hence the minimal polynomial can be found using the quotient group ${\displaystyle {\text{Gal}}(\mathbb {Q} ({\sqrt {2}},{\sqrt {3}})/\mathbb {Q} )/{\text{Gal}}(\mathbb {Q} ({\sqrt {3}})/\mathbb {Q} )}$.

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

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 978-0-486-47417-5