integral basis of quadratic field

Let m be a squarefreeMathworldPlanetmath integer 1. All numbers of the quadratic fieldMathworldPlanetmath (m) may be written in the form

α=j+kml, (1)

where j,k,l are integers with  gcd(j,k,l)=1  and  l>0.  Then α (and its algebraic conjugateα=j-kml) satisfies the equation

x2+px+q= 0, (2)


p=-2jl,q=j2-k2ml2. (3)

We will find out when the number (1) is an algebraic integerMathworldPlanetmath, i.e. when the coefficients p and q are rational integers.

Naturally, p and q are integers always when  l=1.  We suppose now that  l>1.  The latter of the equations (3) says that q can be integer only when


(see divisibility in rings).  Because  gcd(j,k,l)=1,  we have by Euclid’s lemma that  gcd(j,l)m.  Since m is squarefree, we infer that

gcd(j,l)=1. (4)

In order that also p were an integer, the former of the equations (3) implies that  l=2.

So, by the latter of the equations (3),  4j2-k2m, i.e.

k2mj2(mod4). (5)

Since by (4),  gcd(j, 2)=1,  the integer j has to be odd.  In order that (5) would be valid, also k must be odd.  Therefore,  j21(mod4)  and  k21(mod4),  and thus (5) changes to

m1(mod4). (6)

If we conversely assume (6) and that j,k are odd and  l=2, then (5) is true, p,q are integers and accordingly (1) is an algebraic integer.

We have now obtained the following result:

  • When  m1(mod4),  the integers of the field (m) are


    where a,b are arbitrary rational integers;

  • when  m1(mod4),  in to the numbers a+bm, also the numbers


    with j,k arbitrary odd integers, are integers of the field.

Then, it may be easily inferred the

Theorem.  If we denote

ω:={1+m2when m1(mod4),m when m1(mod4),

then any integer of the quadratic field (m) may be expressed in the form


where a and b are uniquely determined rational integers.  Conversely, every number of this form is an integer of the field.  One says that 1 and ω form an integral basis of the field.


  • 1 K. Väisälä: Lukuteorian ja korkeamman algebran alkeet.  Tiedekirjasto No. 17. Kustannusosakeyhtiö Otava, Helsinki (1950).

Title integral basis of quadratic field
Canonical name IntegralBasisOfQuadraticField
Date of creation 2014-02-27 10:24:31
Last modified on 2014-02-27 10:24:31
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 11
Author pahio (2872)
Entry type Derivation
Classification msc 11R04
Synonym canonical basis of quadratic field
Synonym quadratic integers
Related topic PropertiesOfQuadraticEquation
Related topic Gcd
Related topic ExamplesOfRingOfIntegersOfANumberField
Related topic SomethingRelatedToFundamentalUnits
Related topic CanonicalBasis