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Minimal polynomial (field theory)

## Summary

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 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. The set Jα is so named because it is an ideal of F[x]. 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, 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 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 subα from F[x] to E that substitutes α for x, that is, subα(f(x)) = f(α). The kernel of subα, ker(subα), is the set of all polynomials in F[x] that have α as a root. That is, ker(subα) = Jα from above. Since subα is a ring homomorphism, ker(subα) 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(subα) that generates ker(subα). Such a polynomial will have least degree among all non-zero polynomials in ker(subα), 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 Jα of minimal degree n > 0. Since pqJα and deg(pq) < n it follows that pq = 0, i.e. p = q.

## Properties

A minimal polynomial is irreducible. Let E/F be a field extension over F as above, αE, and fF[x] a minimal polynomial for α. Suppose f = gh, where g, hF[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.

#### 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) = x2.

#### 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.

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 ${\displaystyle \alpha ={\sqrt {2}}}$  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.