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Algebraic Integers

The ordinary integers

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Beyond the Ordinary Integers

The ordinary integers

Z={…,−2,−1,0,1,2,… } \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

x2+1=0 x^2+1=0

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

x=±i. x=\pm i.

Similarly, Pell equations lead naturally to expressions involving

D. \sqrt D.

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

Algebraic Numbers

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

anxn+an−1xn−1+⋯+a0=0, a_nx^n+a_{n-1}x^{n-1}+\cdots+a_0=0,

where the coefficients are integers and

an≠0. a_n\ne0.

For example:

2 \sqrt2

is algebraic because it satisfies

x2−2=0. x^2-2=0.

The number

1+52 \frac{1+\sqrt5}{2}

is algebraic because it satisfies

x2−x−1=0. x^2-x-1=0.

The complex number

i i

satisfies

x2+1=0. x^2+1=0.

Thus algebraic numbers extend the ordinary integers and rationals.

Definition of Algebraic Integer

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

xn+an−1xn−1+⋯+a0=0, x^n+a_{n-1}x^{n-1}+\cdots+a_0=0,

with all coefficients in Z\mathbb{Z}.

xn+an−1xn−1+⋯+a0=0 x^n+a_{n-1}x^{n-1}+\cdots+a_0=0

The polynomial must have leading coefficient 11.

For example:

  • 2\sqrt2 is an algebraic integer because
x2−2=0. x^2-2=0.
  • ii is an algebraic integer because
x2+1=0. x^2+1=0.
  • The golden ratio
1+52 \frac{1+\sqrt5}{2}

is an algebraic integer because

x2−x−1=0. x^2-x-1=0.

By contrast,

12 \frac12

is not an algebraic integer. Although it satisfies

2x−1=0, 2x-1=0,

the polynomial is not monic.

Ordinary Integers Inside Algebraic Integers

Every ordinary integer is an algebraic integer.

Indeed, if

n∈Z, n\in\mathbb{Z},

then nn satisfies

x−n=0, x-n=0,

which is monic.

Thus the algebraic integers generalize the usual integers.

The set of all algebraic integers is usually denoted by

Z‾. \overline{\mathbb{Z}}.

Closure Properties

Algebraic integers behave much like ordinary integers.

Theorem. If α\alpha and β\beta are algebraic integers, then so are:

α+β,α−β,αβ. \alpha+\beta, \qquad \alpha-\beta, \qquad \alpha\beta.

Thus algebraic integers form a ring.

For example,

1+2 1+\sqrt2

is an algebraic integer because

(x−1)2−2=0, (x-1)^2-2=0,

which simplifies to

x2−2x−1=0. x^2-2x-1=0.

Similarly,

2⋅3=6 \sqrt2\cdot\sqrt3=\sqrt6

is an algebraic integer because

x2−6=0. 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) (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 DD, one studies the quadratic field

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

Its algebraic integers form a ring usually written

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

Often this ring equals

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

though sometimes extra elements appear.

For example, in

Q(5), \mathbb{Q}(\sqrt5),

the element

1+52 \frac{1+\sqrt5}{2}

is an algebraic integer.

Hence the full ring of integers is larger than

Z[5]. \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 Q\mathbb{Q}.

This polynomial is called the minimal polynomial.

For example:

  • 2\sqrt2 has minimal polynomial
x2−2, x^2-2,
  • the golden ratio has minimal polynomial
x2−x−1. 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,

α=a+bD, \alpha=a+b\sqrt D,

the conjugate is

α‾=a−bD. \overline{\alpha}=a-b\sqrt D.

The norm is

$$ N(\alpha)=\alpha\overline{\alpha}

a^2-Db^2, $$

and the trace is

Tr⁡(α)=α+α‾=2a. \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

Z[−5], \mathbb{Z}[\sqrt{-5}],

one has

6=2⋅3=(1+−5)(1−−5), 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

xn+yn=zn. 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.