# Algebraic Integers

## Beyond the Ordinary Integers

The ordinary integers

$$
\mathbb{Z}=\{\dots,-2,-1,0,1,2,\dots\}
$$

form the basic arithmetic system of number theory. However, many Diophantine equations naturally lead to larger number systems.

For example, the equation

$$
x^2+1=0
$$

has no solution in $\mathbb{Z}$ or $\mathbb{R}$, but it has solutions in the complex numbers:

$$
x=\pm i.
$$

Similarly, Pell equations lead naturally to expressions involving

$$
\sqrt D.
$$

To study arithmetic systematically in such settings, one introduces algebraic integers.

## Algebraic Numbers

A complex number $\alpha$ is called algebraic if it satisfies a polynomial equation

$$
a_nx^n+a_{n-1}x^{n-1}+\cdots+a_0=0,
$$

where the coefficients are integers and

$$
a_n\ne0.
$$

For example:

$$
\sqrt2
$$

is algebraic because it satisfies

$$
x^2-2=0.
$$

The number

$$
\frac{1+\sqrt5}{2}
$$

is algebraic because it satisfies

$$
x^2-x-1=0.
$$

The complex number

$$
i
$$

satisfies

$$
x^2+1=0.
$$

Thus algebraic numbers extend the ordinary integers and rationals.

## Definition of Algebraic Integer

An algebraic number $\alpha$ is called an algebraic integer if it satisfies a monic polynomial equation

$$
x^n+a_{n-1}x^{n-1}+\cdots+a_0=0,
$$

with all coefficients in $\mathbb{Z}$.

$$
x^n+a_{n-1}x^{n-1}+\cdots+a_0=0
$$

The polynomial must have leading coefficient $1$.

For example:

- $\sqrt2$ is an algebraic integer because

$$
x^2-2=0.
$$

- $i$ is an algebraic integer because

$$
x^2+1=0.
$$

- The golden ratio

$$
\frac{1+\sqrt5}{2}
$$

is an algebraic integer because

$$
x^2-x-1=0.
$$

By contrast,

$$
\frac12
$$

is not an algebraic integer. Although it satisfies

$$
2x-1=0,
$$

the polynomial is not monic.

## Ordinary Integers Inside Algebraic Integers

Every ordinary integer is an algebraic integer.

Indeed, if

$$
n\in\mathbb{Z},
$$

then $n$ satisfies

$$
x-n=0,
$$

which is monic.

Thus the algebraic integers generalize the usual integers.

The set of all algebraic integers is usually denoted by

$$
\overline{\mathbb{Z}}.
$$

## Closure Properties

Algebraic integers behave much like ordinary integers.

**Theorem.** If $\alpha$ and $\beta$ are algebraic integers, then so are:

$$
\alpha+\beta,
\qquad
\alpha-\beta,
\qquad
\alpha\beta.
$$

Thus algebraic integers form a ring.

For example,

$$
1+\sqrt2
$$

is an algebraic integer because

$$
(x-1)^2-2=0,
$$

which simplifies to

$$
x^2-2x-1=0.
$$

Similarly,

$$
\sqrt2\cdot\sqrt3=\sqrt6
$$

is an algebraic integer because

$$
x^2-6=0.
$$

These closure properties make algebraic integers suitable for arithmetic.

## Gaussian Integers

One of the simplest examples is the ring of Gaussian integers:

$$
\mathbb{Z}[i]
=
\{a+bi:a,b\in\mathbb{Z}\}.
$$

Every Gaussian integer is an algebraic integer because

$$
(a+bi)
$$

satisfies

$$
(x-(a+bi))(x-(a-bi))
=
x^2-2ax+(a^2+b^2).
$$

All coefficients are integers.

The Gaussian integers extend ordinary arithmetic into the complex plane and play a major role in quadratic reciprocity and sums of squares.

## Quadratic Integer Rings

For a squarefree integer $D$, one studies the quadratic field

$$
\mathbb{Q}(\sqrt D).
$$

Its algebraic integers form a ring usually written

$$
\mathcal O_{\mathbb{Q}(\sqrt D)}.
$$

Often this ring equals

$$
\mathbb{Z}[\sqrt D],
$$

though sometimes extra elements appear.

For example, in

$$
\mathbb{Q}(\sqrt5),
$$

the element

$$
\frac{1+\sqrt5}{2}
$$

is an algebraic integer.

Hence the full ring of integers is larger than

$$
\mathbb{Z}[\sqrt5].
$$

Understanding these rings becomes central in algebraic number theory.

## Minimal Polynomials

Every algebraic number satisfies many polynomial equations. Among them there is a unique monic irreducible polynomial of smallest degree over $\mathbb{Q}$.

This polynomial is called the minimal polynomial.

For example:

- $\sqrt2$ has minimal polynomial

$$
x^2-2,
$$

- the golden ratio has minimal polynomial

$$
x^2-x-1.
$$

An algebraic number is an algebraic integer precisely when its minimal polynomial has integer coefficients.

## Norm and Trace

Algebraic integers possess arithmetic invariants called norm and trace.

For example, in quadratic fields,

$$
\alpha=a+b\sqrt D,
$$

the conjugate is

$$
\overline{\alpha}=a-b\sqrt D.
$$

The norm is

$$
N(\alpha)=\alpha\overline{\alpha}
=
a^2-Db^2,
$$

and the trace is

$$
\operatorname{Tr}(\alpha)=\alpha+\overline{\alpha}=2a.
$$

These quantities generalize familiar arithmetic operations and play a central role in factorization theory.

## Failure of Unique Factorization

Ordinary integers satisfy unique factorization into primes.

Algebraic integers need not.

For example, in

$$
\mathbb{Z}[\sqrt{-5}],
$$

one has

$$
6=2\cdot3=(1+\sqrt{-5})(1-\sqrt{-5}),
$$

and these factorizations are genuinely different.

This failure motivated the introduction of ideals and eventually the development of modern algebraic number theory.

## Historical Perspective

The theory of algebraic integers emerged from attempts to solve Diophantine equations such as Fermat’s equation

$$
x^n+y^n=z^n.
$$

entity["people","Ernst Kummer","German mathematician"] discovered that arithmetic inside cyclotomic fields required new notions of divisibility and factorization.

His work led to ideals, class groups, and the foundations of algebraic number theory.

Algebraic integers thus form the natural arithmetic objects inside algebraic number fields, extending ordinary integers into broader algebraic systems.