# Minimal polynomial (field theory)

In field theory, a branch of mathematics, 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, whose every coefficient is 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 minimal polynomial of *α* is the monic polynomial of least degree among all polynomials in *F*[*x*] having *α* as a root; it exists when *α* is algebraic over *F*, that is, when *f*(*α*) = 0 for some non-zero polynomial *f*(*x*) in *F*[*x*].

### 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*and deg

_{α}*(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

If *F* = **Q**, *E* = **R**, *α* = √2, then the minimal polynomial for *α* is *a*(*x*) = *x*^{2} − 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.

If *α* = √2 + √3, then the minimal polynomial in **Q**[*x*] is *a*(*x*) = *x*^{4} − 10*x*^{2} + 1 = (*x* − √2 − √3)(*x* + √2 − √3)(*x* − √2 + √3)(*x* + √2 + √3).

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.

The minimal polynomials in **Q**[*x*] of roots of unity are the cyclotomic polynomials.

## References

- 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