list of all imaginary quadratic extensions whose ring of integers is a PID

Gauss conjectured that for any Ξ”<0, Δ≑0,1(mod4), then π’žΞ”=1 precisely when


In fact, he believed that as Ξ”β†’-∞,Δ≑0,1(mod4), so does the number of classes of (primitive positive integral binary quadratic) forms with Ξ”.

It is relatively easy to show that the only Δ≑0(mod4) with this property are the ones in this list; that proof is given in an addendum to this article.

However, proving the remainder of Gauss’ hypotheses, regarding the odd values in the list, proved significantly harder. In the first half of the 20th century, Siegel showed that there was at most one such value beyond what Gauss had found. Heegner, Stark, and Baker showed, about 30 years later, that there are in fact no more ([2],[3],[4]).

Thus given an imaginary quadratic extension K, it follows that the ring of integersMathworldPlanetmath of K, denoted π’ͺK, is a PID if and only if the class groupMathworldPlanetmath of K is trivial if and only if there is only one class of primitive quadratic formsMathworldPlanetmath of the appropriate dK if and only if dK is in the set above. So in particular, there are a finite number of imaginary quadratic extensions of β„š whose ring of integers is a PID (and hence a UFD).

The values of Ξ” above that correspond to π’ͺK for some K are:

Ξ”=dK K π’ͺK
-3 β„šβ’(-3) ℀⁒[1+-32]
-4 β„šβ’(-1) ℀⁒[-1]
-7 β„šβ’(-7) ℀⁒[1+-72]
-8 β„šβ’(-2) ℀⁒[-2]
-11 β„šβ’(-11) ℀⁒[1+-112]
-19 β„šβ’(-19) ℀⁒[1+-192]
-43 β„šβ’(-43) ℀⁒[1+-432]
-67 β„šβ’(-67) ℀⁒[1+-672]
-163 β„šβ’(-163) ℀⁒[1+-1632]

We therefore get

Theorem 1.

If d<0, then the class numberMathworldPlanetmath of Q⁒(d) is equal to 1 if and only if

d=-1,-2,-3,-7,-11,-19,-43,-67,Β orΒ -163

(where d=-1,-2 correspond to Ξ”=-4,-8 and otherwise d=Ξ”).

How about the other four values Ξ”=-12,-16,-27,-28? Each of these corresponds to a non-maximal in a quadratic extension (i.e. a proper subring of the ring of algebraic integers). Specifically, we have

Ξ”=-12 ↔ ℀⁒[2⁒1+-32]=℀⁒[-3]⊊π’ͺK⁒ for ⁒K=β„šβ’(-3)
Ξ”=-16 ↔ ℀⁒[2⁒-1]⊊π’ͺK⁒ for ⁒K=β„šβ’(-1)
Ξ”=-27 ↔ ℀⁒[3⁒1+-32]⊊π’ͺK⁒ for ⁒K=β„šβ’(-3)
Ξ”=-28 ↔ ℀⁒[2⁒1+-72]=℀⁒[-7]⊊π’ͺK⁒ for ⁒K=β„šβ’(-7)

Note that this does not mean that these rings are PIDs, since the invertible ideals in an order that is not the entire ring of integers do not include all ideals.


  • 1 Cox,Β D.A. Primes of the Form x2+n⁒y2: Fermat, Class Field Theory, and Complex MultiplicationMathworldPlanetmath, Wiley 1997.
  • 2 Heegner,Β K., Diophantische Analysis und Modulfunktionen, Math. Zeit., 56 (1952), pp. 227-253.
  • 3 Stark,Β H.M., A completePlanetmathPlanetmathPlanetmath determination of the complex quadratic fields with class number one, Mich. Math. J., 14 (1967), pp. 1-27.
  • 4 Baker,Β A., Linear forms in the logarithms of algebraic numbersMathworldPlanetmath, Mathematika, 13 (1966), pp. 204-216.
Title list of all imaginary quadratic extensions whose ring of integers is a PID
Canonical name ListOfAllImaginaryQuadraticExtensionsWhoseRingOfIntegersIsAPID
Date of creation 2013-03-22 16:56:42
Last modified on 2013-03-22 16:56:42
Owner rm50 (10146)
Last modified by rm50 (10146)
Numerical id 7
Author rm50 (10146)
Entry type Result
Classification msc 11E12
Classification msc 11R29
Classification msc 11E16
Related topic EuclideanNumberField
Related topic ImaginaryQuadraticField
Related topic LemmaForImaginaryQuadraticFields