Let K be a quadratic number field with discriminant D. It's well known that a principal factor of K exists if and only if the fundamental unit of K has norm +1. Also I know that a principal factor of K, say t, is a divisor of D. But it's a question for me:" how I can find an explicit method to find of form of t"??
$\begingroup$
$\endgroup$
5
-
1$\begingroup$ Maybe you could remind us what a "principal factor" is. $\endgroup$– Gerry MyersonCommented Oct 15, 2017 at 11:06
-
$\begingroup$ @GerryMyerson Yes, of Course! $\endgroup$– A. MaarefparvarCommented Oct 15, 2017 at 11:30
-
1$\begingroup$ @GerryMyerson A principal factor in K=Q(√m), is an integral principal ideal of K which is composed entirely of ramified primes but is not generated by 1 or m^(1/2). $\endgroup$– A. MaarefparvarCommented Oct 15, 2017 at 11:30
-
$\begingroup$ Thanks. But, you know which primes ramify in a quadratic field, right? mathoverflow.net/questions/137449/… or math.stackexchange.com/questions/376590/… $\endgroup$– Gerry MyersonCommented Oct 15, 2017 at 11:43
-
$\begingroup$ You can easily compute them from the fundamental unit. If no unit is known, check which ramified ideals are principal one by one. But this might tirn out to be difficult without knowing the unit group. $\endgroup$– Franz LemmermeyerCommented Oct 16, 2017 at 21:21
Add a comment
|