Minimal polynomial (field theory)

In field theory, a branch of mathematics, the minimal polynomial of an element $α$ of a field is, roughly speaking, the polynomial of lowest degree having coefficients in the 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, and the type for the remaining coefficients could be integers, rational numbers, real numbers, or others.

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 $J α$ 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 $0 α i = 0$ 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 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.

Uniqueness of monic polynomial[]

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 $L/K$ the minimal polynomial of any $\alpha \in L$ not in $K$ can be computed as

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

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

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

in particular, this implies $2a\in \mathbb {Z}$ and $a^{2}-b^{2}d\in \mathbb {Z}$ . This can be used to determine ${\mathcal {O}}_{\mathbb {Q} ({\sqrt {d}})}$ 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 $\alpha ={\sqrt {2}}$ then the Galois action on ${\sqrt {3}}$ stabilizes $\alpha$ . Hence the minimal polynomial can be found using the quotient group ${\text{Gal}}(\mathbb {Q} ({\sqrt {2}},{\sqrt {3}})/\mathbb {Q} )/{\text{Gal}}(\mathbb {Q} ({\sqrt {3}})/\mathbb {Q} )$ .

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 .

References[]

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