list of all imaginary quadratic extensions whose ring of integers is a PID
Gauss conjectured that for any , , then precisely when
It is relatively easy to show that the only 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 century, Siegel showed that there was at most one such value beyond what Gauss had found. Heegner, Stark, and Baker showed, about years later, that there are in fact no more (,,).
Thus given an imaginary quadratic extension , it follows that the ring of integers of , denoted , is a PID if and only if the class group of is trivial if and only if there is only one class of primitive quadratic forms of the appropriate http://planetmath.org/node/DiscriminantOfANumberFielddiscriminant if and only if is in the set above. So in particular, there are a finite number of imaginary quadratic extensions of whose ring of integers is a PID (and hence a UFD).
The values of above that correspond to for some are:
We therefore get
If , then the class number of is equal to if and only if
(where correspond to and otherwise ).
How about the other four values ? Each of these corresponds to a non-maximal http://planetmath.org/node/OrderInAnAlgebraorder in a quadratic extension (i.e. a proper subring of the ring of algebraic integers). Specifically, we have
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, D.A. Primes of the Form : Fermat, Class Field Theory, and Complex Multiplication, Wiley 1997.
- 2 Heegner, K., Diophantische Analysis und Modulfunktionen, Math. Zeit., 56 (1952), pp. 227-253.
- 3 Stark, H.M., A complete determination of the complex quadratic fields with class number one, Mich. Math. J., 14 (1967), pp. 1-27.
- 4 Baker, A., Linear forms in the logarithms of algebraic numbers, Mathematika, 13 (1966), pp. 204-216.
|Title||list of all imaginary quadratic extensions whose ring of integers is a PID|
|Date of creation||2013-03-22 16:56:42|
|Last modified on||2013-03-22 16:56:42|
|Last modified by||rm50 (10146)|