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In `F33f`_`[number theory`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Number_theory]`_`f, `!quadratic integers`! are a generalization of the usual `F33f`_`[integers`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Integer]`_`f to `F33f`_`[quadratic fields`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Quadratic_field]`_`f. A complex number is called a quadratic integer if it is a `F33f`_`[root`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Root_of_a_polynomial]`_`f of some `F33f`_`[monic polynomial`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Monic_polynomial]`_`f (a `F33f`_`[polynomial`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Polynomial]`_`f whose `F33f`_`[leading coefficient`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Leading_coefficient]`_`f is 1) of degree two whose coefficients are integers, i.e. quadratic integers are `F33f`_`[algebraic integers`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Algebraic_integer]`_`f of degree two. Thus quadratic integers are those complex numbers that are solutions of equations of the form

`*x`*2 + `*bx`* + `*c`* = 0

with b and c (usual) integers. When algebraic integers are considered, the usual integers are often called `*rational integers`*.

Common examples of quadratic integers are the `F33f`_`[square roots`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Square_root]`_`f of rational integers, such as 2 {\\textstyle {\\sqrt {2}}} , and the `F33f`_`[complex number`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Complex_number]`_`f i = − − 1 {\\textstyle i={\\sqrt {-1}}} , which generates the `F33f`_`[Gaussian integers`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Gaussian_integer]`_`f. Another common example is the non-`F33f`_`[real`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Real_number]`_`f cubic `F33f`_`[root of unity`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Root_of_unity]`_`f − − 1 + − − 3 2 {\\textstyle {\\frac {-1+{\\sqrt {-3}}}{2}}} , which generates the `F33f`_`[Eisenstein integers`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Eisenstein_integer]`_`f.

Quadratic integers occur in the solutions of many `F33f`_`[Diophantine equations`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Diophantine_equation]`_`f, such as `F33f`_`[Pell's equations`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Pell's_equation]`_`f, and other questions related to integral `F33f`_`[quadratic forms`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Quadratic_form]`_`f. The study of `!rings of quadratic integers`! is basic for many questions of `F33f`_`[algebraic number theory`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Algebraic_number_theory]`_`f.

>>Contents

• `F0af`_`[History`#history]`_`f

• `F0af`_`[Definition`#definition]`_`f

• `F0af`_`[Explicit representation`#explicit-representation]`_`f

• `F0af`_`[Norm and conjugation`#norm-and-conjugation]`_`f

• `F0af`_`[Quadratic integer rings`#quadratic-integer-rings]`_`f

• `F0af`_`[Units`#units]`_`f

• `F0af`_`[Examples of complex quadratic integer rings`#examples-of-complex-quadratic-integer-rings]`_`f

• `F0af`_`[Examples of real quadratic integer rings`#examples-of-real-quadratic-integer-rings]`_`f

• `F0af`_`[Principal rings of quadratic integers`#principal-rings-of-quadratic-integers]`_`f

• `F0af`_`[Euclidean rings of quadratic integers`#euclidean-rings-of-quadratic-integers]`_`f

• `F0af`_`[Notes`#notes]`_`f

• `F0af`_`[References`#references]`_`f

• `F0af`_`[Further reading`#further-reading]`_`f

-─

>>History

Medieval `F33f`_`[Indian mathematicians`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Indian_mathematics]`_`f had already discovered a multiplication of quadratic integers of the same `F33f`_`[discriminant`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Discriminant]`_`f D, which allowed them to solve some cases of `F33f`_`[Pell's equation`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Pell's_equation]`_`f.

The characterization given in `*`F33f`_`[§ Explicit representation`#explicit-representation]`_`f`* of the quadratic integers was first given by `F33f`_`[Richard Dedekind`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Richard_Dedekind]`_`f in 1871.`:cite-ref-footnotededekind1871supplement-x-p-447-1-0[`F5bf`_`[1`#cite-note-footnotededekind1871supplement-x-p-447-1]`_`f]`:cite-ref-footnotebourbaki199499-2-0[`F5bf`_`[2`#cite-note-footnotebourbaki199499-2]`_`f]

>>Definition

A `!quadratic integer`! is an `F33f`_`[algebraic integer`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Algebraic_integer]`_`f of degree two. More explicitly, it is a `F33f`_`[complex number`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Complex_number]`_`f x = ( − − b ± ± b 2 − − 4 c ) / 2 {\\displaystyle x=(-b\\pm {\\sqrt {b^{2}-4c}})/2} , which solves an equation of the form `*x`*2 + `*bx`* + `*c`* = 0, with `*b`* and `*c`* `F33f`_`[integers`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Integer]`_`f. Each quadratic integer that is not an integer is not `F33f`_`[rational`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Rational_number]`_`f – namely, it's a real `F33f`_`[irrational number`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Irrational_number]`_`f if `*b`*2 − 4`*c`* > 0 and non-real if `*b`*2 − 4`*c`* < 0 – and lies in a uniquely determined `F33f`_`[quadratic field`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Quadratic_field]`_`f Q ( D ) {\\displaystyle \\mathbb {Q} ({\\sqrt {D}}\\,)} , the extension of Q {\\displaystyle \\mathbb {Q} } generated by the square root of the unique `F33f`_`[square-free integer`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Square-free_integer]`_`f D that satisfies `*b`*2 − 4`*c`* = `*De`*2 for some integer `*e`*. If `*D`* is positive, the quadratic integer is real. If `*D`* < 0, it is `*imaginary`* (that is, complex and non-real).

The quadratic integers (including the ordinary integers) that belong to a quadratic field Q ( D ) {\\displaystyle \\mathbb {Q} ({\\sqrt {D}}\\,)} form an `F33f`_`[integral domain`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Integral_domain]`_`f called the `*ring of integers of`* Q ( D ) . {\\displaystyle \\mathbb {Q} ({\\sqrt {D}}\\,).}

Although the quadratic integers belonging to a given quadratic field form a `F33f`_`[ring`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Ring_(mathematics)]`_`f, the set of `*all`* quadratic integers is not a ring because it is not closed under addition or multiplication. For example, 1 + 2 {\\displaystyle 1+{\\sqrt {2}}} and 3 {\\displaystyle {\\sqrt {3}}} are quadratic integers, but 1 + 2 + 3 {\\displaystyle 1+{\\sqrt {2}}+{\\sqrt {3}}} and ( 1 + 2 ) ⋅ ⋅ 3 {\\displaystyle (1+{\\sqrt {2}})\\cdot {\\sqrt {3}}} are not, as their `F33f`_`[minimal polynomials`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Minimal_polynomial_(field_theory)]`_`f have `F33f`_`[degree`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Degree_of_a_polynomial]`_`f four.

>>Explicit representation

Here and in the following, the quadratic integers that are considered belong to a `F33f`_`[quadratic field`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Quadratic_field]`_`f Q ( D ) , {\\displaystyle \\mathbb {Q} ({\\sqrt {D}}\\,),} where D is a square-free integer. This does not restrict the generality, as the equality a 2 D = a D {\\textstyle {\\sqrt {a^{2}D}}=a{\\sqrt {D}}} (for any positive integer a) implies Q ( D ) = Q ( a 2 D ) . {\\textstyle \\mathbb {Q} ({\\sqrt {D}}\\,)=\\mathbb {Q} ({\\sqrt {a^{2}D}}\\,).}

An element x of Q ( D ) {\\textstyle \\mathbb {Q} ({\\sqrt {D}}\\,)} is a quadratic integer if and only if there are two integers a and b such that either

x = a + b D , {\\displaystyle x=a+b{\\sqrt {D}},}

or, if `*D`* − 1 is a multiple of 4

x = a 2 + b 2 D , {\\displaystyle x={\\frac {a}{2}}+{\\frac {b}{2}}{\\sqrt {D}},} with a and b both `F33f`_`[odd`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Parity_(mathematics)]`_`f.

In other words, every quadratic integer may be written `*a`* + `*ωb`* , where a and b are integers, and where ω is defined by

ω ω = { D if D ≡ ≡ 2 , 3 ( mod 4 ) 1 + D 2 if D ≡ ≡ 1 ( mod 4 ) {\\displaystyle \\omega ={\\begin{cases}{\\sqrt {D}}&{\\mbox{if }}D\\equiv 2,3{\\pmod {4}}\\\\{{1+{\\sqrt {D}}} \\over 2}&{\\mbox{if }}D\\equiv 1{\\pmod {4}}\\end{cases}}}

(as D has been supposed square-free the case D ≡ ≡ 0 ( mod 4 ) {\\textstyle D\\equiv 0{\\pmod {4}}} is impossible, since it would imply that D is divisible by the `F33f`_`[square`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Square_number]`_`f 4).`:cite-ref-3[`F5bf`_`[3`#cite-note-3]`_`f]

>>Norm and conjugation

A quadratic integer in Q ( D ) {\\displaystyle \\mathbb {Q} ({\\sqrt {D}}\\,)} may be written

a + b D {\\textstyle a+b{\\sqrt {D}}} ,

where a and b are either both integers, or, only if `*D`* ≡ 1 (mod 4), both `F33f`_`[halves of odd integers`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Half-integer]`_`f. The `!norm`! of such a quadratic integer is

N ( a + b D ) = a 2 − − D b 2 . {\\textstyle N(a+b{\\sqrt {D}})=a^{2}-Db^{2}.}

The norm of a quadratic integer is always an integer. If `*D`* < 0, the norm of a quadratic integer is the square of its `F33f`_`[absolute value`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Absolute_value]`_`f as a complex number (this is false if D > 0 {\\textstyle D>0} ). The norm is a `F33f`_`[completely multiplicative function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Completely_multiplicative_function]`_`f, which means that the norm of a product of quadratic integers is always the product of their norms.

Every quadratic integer a + b D {\\textstyle a+b{\\sqrt {D}}} has a `!conjugate`!

a + b D ¯ ¯ = a − − b D . {\\textstyle {\\overline {a+b{\\sqrt {D}}}}=a-b{\\sqrt {D}}.}

A quadratic integer has the same norm as its conjugate, and this norm is the product of the quadratic integer and its conjugate. The conjugate of a sum or a product of quadratic integers is the sum or the product (respectively) of the conjugates. This means that the conjugation is an `F33f`_`[automorphism`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Automorphism]`_`f of the ring of the integers of Q ( D ) {\\displaystyle \\mathbb {Q} ({\\sqrt {D}}\\,)} – see `*`F33f`_`[§ Quadratic integer rings`#quadratic-integer-rings]`_`f`*, below.

>>Quadratic integer rings

Every `F33f`_`[square-free integer`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Square-free_integer]`_`f (different from 0 and 1) D defines a `!quadratic integer ring`!, which is the `F33f`_`[integral domain`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Integral_domain]`_`f consisting of the `F33f`_`[algebraic integers`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Algebraic_integer]`_`f contained in Q ( D ) . {\\displaystyle \\mathbb {Q} ({\\sqrt {D}}\\,).} It is the set Z [ ω ω ] = { a + ω ω b : a , b ∈ ∈ Z } {\\displaystyle \\mathbb {Z} [\\omega ]=\\{a+\\omega b:a,b\\in \\mathbb {Z} \\}} where ω ω = 1 + D 2 {\\displaystyle \\omega ={\\tfrac {1+{\\sqrt {D}}}{2}}} if `*D`* = 4`*k`* + 1, and `*ω`* = √`*D`* otherwise. It is often denoted O Q ( D ) {\\displaystyle {\\mathcal {O}}_{\\mathbb {Q} ({\\sqrt {D}}\\,)}} , because it is the `F33f`_`[ring of integers`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Ring_of_integers]`_`f of Q ( D ) {\\displaystyle \\mathbb {Q} ({\\sqrt {D}}\\,)} , which is the `F33f`_`[integral closure`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Integral_closure]`_`f of Z {\\displaystyle \\mathbb {Z} } in Q ( D ) . {\\displaystyle \\mathbb {Q} ({\\sqrt {D}}\\,).} The ring Z [ ω ω ] {\\displaystyle \\mathbb {Z} [\\omega ]} consists of all `F33f`_`[roots`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Root_of_a_polynomial]`_`f of all equations `*x`*2 + `*Bx`* + `*C`* = 0 whose `F33f`_`[discriminant`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Discriminant]`_`f `*B`*2 − 4`*C`* is the product of D by the square of an integer. In particular √`*D`* belongs to Z [ ω ω ] {\\displaystyle \\mathbb {Z} [\\omega ]} , being a root of the equation `*x`*2 − `*D`* = 0, which has 4`*D`* as its discriminant.

The square root of any integer is a quadratic integer, as every integer can be written `*n`* = `*m`*2`*D`*, where D is a square-free integer, and its square root is a root of `*x`*2 − `*m`*2`*D`* = 0.

The `F33f`_`[fundamental theorem of arithmetic`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Fundamental_theorem_of_arithmetic]`_`f is not true in many rings of quadratic integers. However, there is a unique factorization for `F33f`_`[ideals`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Ideal_(ring_theory)]`_`f, which is expressed by the fact that every ring of algebraic integers is a `F33f`_`[Dedekind domain`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Dedekind_domain]`_`f. Being the simplest examples of algebraic integers, quadratic integers are commonly the starting examples of most studies of `F33f`_`[algebraic number theory`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Algebraic_number_theory]`_`f.`:cite-ref-footnoteartinch-13-4-0[`F5bf`_`[4`#cite-note-footnoteartinch-13-4]`_`f]

The quadratic integer rings divide in two classes depending on the sign of D. If `*D`* > 0, all elements of O Q ( D ) {\\displaystyle {\\mathcal {O}}_{\\mathbb {Q} ({\\sqrt {D}}\\,)}} are real, and the ring is a `*real quadratic integer ring`*. If `*D`* < 0, the only real elements of O Q ( D ) {\\displaystyle {\\mathcal {O}}_{\\mathbb {Q} ({\\sqrt {D}}\\,)}} are the ordinary integers, and the ring is a `*complex quadratic integer ring`*.

For real quadratic integer rings, the `F33f`_`[class number`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Class_number_(number_theory)]`_`f – which measures the failure of unique factorization – is given in OEIS A003649; for the imaginary case, they are given in OEIS A000924.

>>>Units

A quadratic integer is a `F33f`_`[unit`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Unit_(ring_theory)]`_`f in the ring of the integers of Q ( D ) {\\displaystyle \\mathbb {Q} ({\\sqrt {D}}\\,)} if and only if its norm is 1 or −1. In the first case its `F33f`_`[multiplicative inverse`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Multiplicative_inverse]`_`f is its conjugate. It is the `F33f`_`[negation`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Additive_inverse]`_`f of its conjugate in the second case.

If `*D`* < 0, the ring of the integers of Q ( D ) {\\displaystyle \\mathbb {Q} ({\\sqrt {D}}\\,)} has at most six units. In the case of the `F33f`_`[Gaussian integers`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Gaussian_integer]`_`f (`*D`* = −1), the four units are 1 , − − 1 , − − 1 , − − − − 1 {\\textstyle 1,-1,{\\sqrt {-1}},-{\\sqrt {-1}}} . In the case of the `F33f`_`[Eisenstein integers`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Eisenstein_integer]`_`f (`*D`* = −3), the six units are ± ± 1 , ± ± 1 ± ± − − 3 2 {\\textstyle \\pm 1,{\\frac {\\pm 1\\pm {\\sqrt {-3}}}{2}}} . For all other negative D, there are only two units, which are 1 and −1.

If `*D`* > 0, the ring of the integers of Q ( D ) {\\displaystyle \\mathbb {Q} ({\\sqrt {D}}\\,)} has infinitely many units that are equal to ± `*u`*`*i`*, where i is an arbitrary integer, and u is a particular unit called a `*`F33f`_`[fundamental unit`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Fundamental_unit_(number_theory)]`_`f`*. Given a fundamental unit u, there are three other fundamental units, its conjugate u ¯ ¯ , {\\displaystyle {\\overline {u}},} and also − − u {\\displaystyle -u} and − − u ¯ ¯ . {\\displaystyle -{\\overline {u}}.} Commonly, one calls "`*the`* fundamental unit" the unique one which has an absolute value greater than 1 (as a real number). It is the unique fundamental unit that may be written as `*a`* + `*b`*√`*D`*, with a and b positive (integers or halves of integers).

The fundamental units for the 10 smallest positive square-free D are 1 + 2 {\\textstyle 1+{\\sqrt {2}}} , 2 + 3 {\\textstyle 2+{\\sqrt {3}}} , 1 + 5 2 {\\textstyle {\\frac {1+{\\sqrt {5}}}{2}}} (the `F33f`_`[golden ratio`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Golden_ratio]`_`f), 5 + 2 6 {\\textstyle 5+2{\\sqrt {6}}} , 8 + 3 7 {\\textstyle 8+3{\\sqrt {7}}} , 3 + 10 {\\textstyle 3+{\\sqrt {10}}} , 10 + 3 11 {\\textstyle 10+3{\\sqrt {11}}} , 3 + 13 2 {\\textstyle {\\frac {3+{\\sqrt {13}}}{2}}} , 15 + 4 14 {\\textstyle 15+4{\\sqrt {14}}} , 4 + 15 {\\textstyle 4+{\\sqrt {15}}} . For larger D, the `F33f`_`[coefficients`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Coefficient]`_`f of the fundamental unit may be very large. For example, for `*D`* = 19, 31, 43, the fundamental units are respectively 170 + 39 19 {\\textstyle 170+39{\\sqrt {19}}} , 1520 + 273 31 {\\textstyle 1520+273{\\sqrt {31}}} and 3482 + 531 43 {\\textstyle 3482+531{\\sqrt {43}}} .

>>>Examples of complex quadratic integer rings

For D < 0, ω is a complex (`F33f`_`[imaginary`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Imaginary_number]`_`f or otherwise non-real) number. Therefore, it is natural to treat a quadratic integer ring as a set of algebraic `F33f`_`[complex numbers`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Complex_number]`_`f.

• A classic example is Z [ − − 1 ] {\\displaystyle \\mathbf {Z} [{\\sqrt {-1}}\\,]} , the `F33f`_`[Gaussian integers`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Gaussian_integer]`_`f, which was introduced by `F33f`_`[Carl Gauss`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Carl_Gauss]`_`f around 1800 to state his biquadratic reciprocity law.`:cite-ref-footnotedummitfoote2004229-5-0[`F5bf`_`[5`#cite-note-footnotedummitfoote2004229-5]`_`f]

• The elements in O Q ( − − 3 ) = Z [ 1 2 ( 1 + − − 3 ) ] {\\displaystyle {\\mathcal {O}}_{\\mathbf {Q} ({\\sqrt {-3}}\\,)}=\\mathbf {Z} \\left[{\\tfrac {1}{2}}{\\bigl (}1+{\\sqrt {-3}}~\\!{\\bigr )}\\right]} are called `F33f`_`[Eisenstein integers`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Eisenstein_integer]`_`f.

• The elements in O Q ( − − 7 ) = Z [ 1 2 ( 1 + − − 7 ) ] {\\displaystyle {\\mathcal {O}}_{\\mathbf {Q} ({\\sqrt {-7}}\\,)}=\\mathbf {Z} \\left[{\\tfrac {1}{2}}{\\bigl (}1+{\\sqrt {-7}}~\\!{\\bigr )}\\right]} are called `F33f`_`[Kleinian integers`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Kleinian_integer]`_`f

The first two rings mentioned above are rings of integers of `F33f`_`[cyclotomic fields`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Cyclotomic_field]`_`f `!Q`!(`*ζ`*4) and `!Q`!(`*ζ`*3) correspondingly. In contrast, Z [ − − 3 ] {\\displaystyle \\mathbf {Z} {\\bigl [}{\\sqrt {-3}}~\\!{\\bigr ]}} is not even a `F33f`_`[Dedekind domain`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Dedekind_domain]`_`f.

All the above examples are `F33f`_`[principal ideal rings`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Principal_ideal_ring]`_`f and also `F33f`_`[Euclidean domains`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Euclidean_domain]`_`f for the norm. This is not the case for

O Q ( − − 5 ) = Z [ − − 5 ] , {\\displaystyle {\\mathcal {O}}_{\\mathbf {Q} ({\\sqrt {-5}}\\,)}=\\mathbf {Z} \\left[{\\sqrt {-5}}\\,\\right],}

which is not even a `F33f`_`[unique factorization domain`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Unique_factorization_domain]`_`f. This can be shown as follows.

In O Q ( − − 5 ) , {\\displaystyle {\\mathcal {O}}_{\\mathbf {Q} ({\\sqrt {-5}}\\,)},} we have

9 = 3 ⋅ ⋅ 3 = ( 2 + − − 5 ) ( 2 − − − − 5 ) . {\\displaystyle 9=3\\cdot 3=(2+{\\sqrt {-5}})(2-{\\sqrt {-5}}).}

The factors 3, 2 + − − 5 {\\displaystyle 2+{\\sqrt {-5}}} and 2 − − − − 5 {\\displaystyle 2-{\\sqrt {-5}}} are `F33f`_`[irreducible`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Irreducible_element]`_`f, as they have all a norm of 9, and if they were not irreducible, they would have a factor of norm 3, which is impossible, the norm of an element different of ±1 being at least 4. Thus the factorization of 9 into irreducible factors is not unique.

The `F33f`_`[ideals`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Ideal_(ring_theory)]`_`f ⟨ ⟨ 3 , 1 + − − 5 ⟩ ⟩ {\\displaystyle \\langle 3,1+{\\sqrt {-5}}\\,\\rangle } and ⟨ ⟨ 3 , 1 − − − − 5 ⟩ ⟩ {\\displaystyle \\langle 3,1-{\\sqrt {-5}}\\,\\rangle } are not `F33f`_`[principal`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Principal_ideal]`_`f, as a simple computation shows that their product is the ideal generated by 3, and, if they were principal, this would imply that 3 would not be irreducible.

>>>Examples of real quadratic integer rings

For `*D`* > 0, ω is a positive irrational `F33f`_`[real number`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Real_number]`_`f, and the corresponding quadratic integer ring is a set of algebraic real numbers. The solutions of the `F33f`_`[Pell's equation`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Pell's_equation]`_`f `*X`* 2 − `*DY`* 2 = 1, a `F33f`_`[Diophantine equation`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Diophantine_equation]`_`f that has been widely studied, are the `F33f`_`[units`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Unit_(ring_theory)]`_`f of these rings, for `*D`* ≡ 2, 3 (mod 4).

• For `*D`* = 5, `*ω`* = ⁠1/2⁠(1+√5) is the `F33f`_`[golden ratio`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Golden_ratio]`_`f. This `F33f`_`[golden integer`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Golden_integer]`_`f ring was studied by `F33f`_`[Peter Gustav Lejeune Dirichlet`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Peter_Gustav_Lejeune_Dirichlet]`_`f. Its units have the form ±`*ω`*`*n`*, where n is an arbitrary integer. This ring also arises from studying 5-fold `F33f`_`[rotational symmetry`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Rotational_symmetry]`_`f on Euclidean plane, for example, `F33f`_`[Penrose tilings`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Penrose_tiling]`_`f.`:cite-ref-footnotede-bruijn1981-6-0[`F5bf`_`[6`#cite-note-footnotede-bruijn1981-6]`_`f]

• Indian mathematician `F33f`_`[Brahmagupta`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Brahmagupta]`_`f treated the Pell's equation `*X`*2 − 61`*Y`*2 = 1, corresponding to the ring is `!Z`![√61]. Some results were presented to European community by `F33f`_`[Pierre Fermat`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Pierre_Fermat]`_`f in 1657.

>>>Principal rings of quadratic integers

The `F33f`_`[unique factorization`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Unique_factorization]`_`f property is not always verified for rings of quadratic integers, as seen above for the case of `!Z`![√−5]. However, as for every `F33f`_`[Dedekind domain`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Dedekind_domain]`_`f, a ring of quadratic integers is a `F33f`_`[unique factorization domain`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Unique_factorization_domain]`_`f if and only if it is a `F33f`_`[principal ideal domain`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Principal_ideal_domain]`_`f. This occurs if and only if the `F33f`_`[class number`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Ideal_class_group]`_`f of the corresponding `F33f`_`[quadratic field`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Quadratic_field]`_`f is one.

The imaginary rings of quadratic integers that are principal ideal rings have been completely determined. These are O Q ( D ) {\\displaystyle {\\mathcal {O}}_{\\mathbf {Q} ({\\sqrt {D}}\\,)}} for

`*D`* = −1, −2, −3, −7, −11, −19, −43, −67, −163.

This result was first `F33f`_`[conjectured`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Conjecture]`_`f by `F33f`_`[Gauss`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Gauss]`_`f and `F33f`_`[proven`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Mathematical_proof]`_`f by `F33f`_`[Kurt Heegner`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Kurt_Heegner]`_`f, although Heegner's proof was not believed until `F33f`_`[Harold Stark`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Harold_Stark]`_`f gave a later proof in 1967 (see `*`F33f`_`[Stark–Heegner theorem`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Stark–Heegner_theorem]`_`f`*). This is a special case of the famous `F33f`_`[class number problem`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Class_number_problem]`_`f.

There are many known positive integers `*D`* > 0, for which the ring of quadratic integers is a principal ideal ring. However, the complete list is not known; it is not even known if the number of these principal ideal rings is finite or not.

>>>Euclidean rings of quadratic integers

When a ring of quadratic integers is a principal ideal domain, it is interesting to know whether it is a `F33f`_`[Euclidean domain`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Euclidean_domain]`_`f. This problem has been completely solved as follows.

Equipped with the norm N ( a + b D ) = | a 2 − − D b 2 | {\\displaystyle N(a+b{\\sqrt {D}}\\,)=|a^{2}-Db^{2}|} as a `F33f`_`[Euclidean function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Euclidean_function]`_`f, O Q ( D ) {\\displaystyle {\\mathcal {O}}_{\\mathbf {Q} ({\\sqrt {D}}\\,)}} is a Euclidean domain for negative D when

`*D`* = −1, −2, −3, −7, −11,`:cite-ref-footnotedummitfoote2004272-7-0[`F5bf`_`[7`#cite-note-footnotedummitfoote2004272-7]`_`f]

and, for positive D, when

`*D`* = 2, 3, 5, 6, 7, 11, 13, 17, 19, 21, 29, 33, 37, 41, 57, 73 (sequence A048981 in the `F33f`_`[OEIS`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=On-Line_Encyclopedia_of_Integer_Sequences]`_`f).

There is no other ring of quadratic integers that is Euclidean with the norm as a Euclidean function.`:cite-ref-footnoteleveque2002ii-57-81-8-0[`F5bf`_`[8`#cite-note-footnoteleveque2002ii-57-81-8]`_`f] For negative D, a ring of quadratic integers is Euclidean if and only if the norm is a `F33f`_`[Euclidean function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Euclidean_function]`_`f for it. It follows that, for

`*D`* = −19, −43, −67, −163,

the four corresponding rings of quadratic integers are among the rare known examples of principal ideal domains that are not Euclidean domains.

On the other hand, the `F33f`_`[generalized Riemann hypothesis`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Generalized_Riemann_hypothesis]`_`f implies that a ring of `*real`* quadratic integers that is a principal ideal domain is also a Euclidean domain for some Euclidean function, which can indeed differ from the usual norm.`:cite-ref-9[`F5bf`_`[9`#cite-note-9]`_`f] The values `*D`* = 14, 69 were the first for which the ring of quadratic integers was proven to be Euclidean, but not norm-Euclidean.`:cite-ref-footnoteharper2004-10-0[`F5bf`_`[10`#cite-note-footnoteharper2004-10]`_`f]`:cite-ref-footnoteclark1994-11-0[`F5bf`_`[11`#cite-note-footnoteclark1994-11]`_`f]

>>Notes

`:cite-note-footnotededekind1871supplement-x-p-447-1`!1.`! `F0af`_`[↑`#cite-ref-footnotededekind1871supplement-x-p-447-1-0]`_`f `F33f`_`[Dedekind 1871`#citerefdedekind1871]`_`f, Supplement X, p. 447

`:cite-note-footnotebourbaki199499-2`!2.`! `F0af`_`[↑`#cite-ref-footnotebourbaki199499-2-0]`_`f `F33f`_`[Bourbaki 1994`#citerefbourbaki1994]`_`f, p. 99

`:cite-note-3`!3.`! `F0af`_`[↑`#cite-ref-3]`_`f "Why is quadratic integer ring defined in that way?". `*math.stackexchange.com`*. Retrieved 2016-12-31.

`:cite-note-footnoteartinch-13-4`!4.`! `F0af`_`[↑`#cite-ref-footnoteartinch-13-4-0]`_`f `F33f`_`[Artin`#citerefartin]`_`f, Ch 13

`:cite-note-footnotedummitfoote2004229-5`!5.`! `F0af`_`[↑`#cite-ref-footnotedummitfoote2004229-5-0]`_`f `F33f`_`[Dummit & Foote 2004`#citerefdummitfoote2004]`_`f, p. 229

`:cite-note-footnotede-bruijn1981-6`!6.`! `F0af`_`[↑`#cite-ref-footnotede-bruijn1981-6-0]`_`f `F33f`_`[de Bruijn 1981`#citerefde-bruijn1981]`_`f

`:cite-note-footnotedummitfoote2004272-7`!7.`! `F0af`_`[↑`#cite-ref-footnotedummitfoote2004272-7-0]`_`f `F33f`_`[Dummit & Foote 2004`#citerefdummitfoote2004]`_`f, p. 272

`:cite-note-footnoteleveque2002ii-57-81-8`!8.`! `F0af`_`[↑`#cite-ref-footnoteleveque2002ii-57-81-8-0]`_`f `F33f`_`[LeVeque 2002`#citerefleveque2002]`_`f, pp. II:57, 81

`:cite-note-9`!9.`! `F0af`_`[↑`#cite-ref-9]`_`f P. Weinberger, `*On Euclidean rings of algebraic integers`*. In: Analytic Number Theory (St. Louis, 1972), Proc. Sympos. Pure Math. 24(1973), 321–332.

`:cite-note-footnoteharper2004-10`!10.`! `F0af`_`[↑`#cite-ref-footnoteharper2004-10-0]`_`f `F33f`_`[Harper 2004`#citerefharper2004]`_`f

`:cite-note-footnoteclark1994-11`!11.`! `F0af`_`[↑`#cite-ref-footnoteclark1994-11-0]`_`f `F33f`_`[Clark 1994`#citerefclark1994]`_`f

>>References

• `:citerefartin`aArtin, M, `*Algebra`* (2nd ed.)

• `:citerefbourbaki1994`a`F33f`_`[Bourbaki, Nicolas`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Nicolas_Bourbaki]`_`f (1994). `*Elements of the history of mathematics`*. Translated by `F33f`_`[Meldrum, John`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=John_D._P._Meldrum]`_`f. Berlin: Springer-Verlag. `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 978-3-540-64767-6. `F33f`_`[MR`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=MR_(identifier)]`_`f 1290116.

• `:citerefclark1994`aClark, David A. (1994), "A quadratic field which is Euclidean but not norm-Euclidean" (PDF), `*Manuscripta Mathematica`*, `!83`!: 327–330, `F33f`_`[doi`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Doi_(identifier)]`_`f:10.1007/BF02567617, archived from the original (PDF) on 2015-01-29

• `:citerefde-bruijn1981`ade Bruijn, N. G. (1981), "Algebraic theory of Penrose's non-periodic tilings of the plane, I, II" (PDF), `*Indagationes Mathematicae`*, `!43`! (1): 39–66

• `:citerefdedekind1871`a`F33f`_`[Dedekind, Richard`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Richard_Dedekind]`_`f (1871), `*Vorlesungen über Zahlentheorie von P.G. Lejeune Dirichlet`* (2nd ed.), Vieweg, retrieved 2009-08-05

• `:citerefdummitfoote2004`aDummit, D. S.; Foote, R. M. (2004), `*Abstract Algebra`* (3rd ed.)

• `:citerefharper2004`aHarper, M. (2004), " Z [ 14 ] {\\displaystyle \\mathbb {Z} [{\\sqrt {14}}]} `*is Euclidean`*", `*Can. J. Math. 56`*, `!56`!: 55–70, `F33f`_`[doi`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Doi_(identifier)]`_`f:10.4153/CJM-2004-003-9

• `:citerefleveque2002`a`F33f`_`[LeVeque, William J.`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=William_J._LeVeque]`_`f (2002) [1956]. `*Topics in Number Theory, Volumes I and II`*. New York: Dover Publications. `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 978-0-486-42539-9. `F33f`_`[Zbl`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Zbl_(identifier)]`_`f 1009.11001.

>>Further reading

• J.S. Milne. `*Algebraic Number Theory`*, Version 3.01, September 28, 2008. online lecture notes

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