Minimal Polynomials

Algebraic Dependence Over $\mathbb{Q}$

An algebraic number is a complex number that satisfies some nonzero polynomial equation with rational coefficients. Thus $\alpha\in\mathbb{C}$ is algebraic if there exists a nonzero polynomial

f(x)\in\mathbb{Q}[x]

such that

f(\alpha)=0.

Among all such polynomials, one is distinguished: the monic polynomial of least degree. This polynomial is called the minimal polynomial of $\alpha$ over $\mathbb{Q}$ .

The minimal polynomial records the simplest exact algebraic relation satisfied by $\alpha$ .

Definition

Let $\alpha$ be algebraic over $\mathbb{Q}$ . The minimal polynomial of $\alpha$ over $\mathbb{Q}$ is the unique monic irreducible polynomial

m_\alpha(x)\in\mathbb{Q}[x]

such that

m_\alpha(\alpha)=0.

Here monic means that the leading coefficient is $1$ , and irreducible means that the polynomial cannot be factored nontrivially in $\mathbb{Q}[x]$ .

For example, $\sqrt2$ satisfies

x^2-2=0.

The polynomial $x^2-2$ is monic and irreducible over $\mathbb{Q}$ , so it is the minimal polynomial of $\sqrt2$ .

First Examples

The number $i$ satisfies

x^2+1=0.

Since $x^2+1$ has no rational root, it is irreducible over $\mathbb{Q}$ . Hence

m_i(x)=x^2+1.

The number

\frac{1+\sqrt5}{2}

satisfies

x^2-x-1=0.

This polynomial is monic and irreducible over $\mathbb{Q}$ , so it is the minimal polynomial.

By contrast, the rational number $3/4$ has minimal polynomial

x-\frac34.

If one wants integer coefficients, one may write

4x-3,

but the monic minimal polynomial over $\mathbb{Q}$ is $x-3/4$ .

Uniqueness

The minimal polynomial is unique.

Suppose $f(x)$ and $g(x)$ are both monic irreducible polynomials in $\mathbb{Q}[x]$ with

f(\alpha)=0, \qquad g(\alpha)=0.

Since $f$ is irreducible and $g(\alpha)=0$ , the polynomial $f$ must divide $g$ . Similarly, $g$ must divide $f$ . Hence they have the same degree and differ only by a nonzero constant factor.

Because both are monic, that constant factor is $1$ . Therefore

f(x)=g(x).

This proves uniqueness.

Divisibility Property

The minimal polynomial divides every polynomial that vanishes at $\alpha$ .

h(x)\in\mathbb{Q}[x]

and

h(\alpha)=0,

then

m_\alpha(x)\mid h(x)

in $\mathbb{Q}[x]$ .

This property makes the minimal polynomial the fundamental algebraic relation of $\alpha$ .

For example, $\sqrt2$ also satisfies

x^4-4=0.

But

x^4-4=(x^2-2)(x^2+2),

and its minimal polynomial $x^2-2$ appears as a factor.

Degree of an Algebraic Number

The degree of an algebraic number $\alpha$ is the degree of its minimal polynomial.

Thus

\deg(\sqrt2)=2, \qquad \deg(i)=2, \qquad \deg\left(\frac{1+\sqrt5}{2}\right)=2.

Every rational number has degree $1$ , since its minimal polynomial is linear.

The degree measures the algebraic complexity of the number over $\mathbb{Q}$ . Larger degree means the number requires a higher-degree equation to describe it exactly.

Algebraic Integers and Minimal Polynomials

Minimal polynomials give a clean criterion for algebraic integers.

Theorem. An algebraic number $\alpha$ is an algebraic integer if and only if its minimal polynomial over $\mathbb{Q}$ lies in

\mathbb{Z}[x]

and is monic.

For example,

\sqrt2

is an algebraic integer because its minimal polynomial is

x^2-2.

But

\frac12

is not an algebraic integer because its minimal polynomial is

x-\frac12,

which does not have integer coefficients.

This criterion explains why algebraic integers generalize ordinary integers rather than rational numbers.

Conjugates

The roots of the minimal polynomial are called the conjugates of $\alpha$ .

For example, the minimal polynomial of $\sqrt2$ is

x^2-2,

whose roots are

\sqrt2 \quad\text{and}\quad -\sqrt2.

Thus the conjugates of $\sqrt2$ are $\sqrt2$ and $-\sqrt2$ .

For the golden ratio

\varphi=\frac{1+\sqrt5}{2},

the conjugate is

\frac{1-\sqrt5}{2}.

Conjugates play a central role in defining norm, trace, discriminant, and embeddings of number fields.

Minimal Polynomials and Number Fields

If $\alpha$ is algebraic, then the field

\mathbb{Q}(\alpha)

is generated by adjoining $\alpha$ to $\mathbb{Q}$ . Its degree over $\mathbb{Q}$ equals the degree of the minimal polynomial:

[\mathbb{Q}(\alpha):\mathbb{Q}] = \deg m_\alpha.

For example,

[\mathbb{Q}(\sqrt2):\mathbb{Q}]=2.

The field consists of all expressions

a+b\sqrt2, \qquad a,b\in\mathbb{Q}.

Thus minimal polynomials connect individual algebraic numbers with the fields they generate.

Arithmetic Meaning

Minimal polynomials are not merely equations. They encode arithmetic data.

From the minimal polynomial, one can read:

the degree of the algebraic number,

the conjugates,

whether the number is an algebraic integer,

the field generated by the number,

and, in many cases, norm and trace.

For an algebraic integer, the minimal polynomial plays the role of an exact arithmetic fingerprint. It is the smallest polynomial identity that forces the number to exist.