Timeline for Number fields with same discriminant and regulator?
Current License: CC BY-SA 2.5
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 23, 2010 at 20:01 | comment | added | KConrad | Letting u = (1+sqrt(5))/2, the unit group of Q(zeta_5) is mu_{10} x <u> and the unit group of Q(sqrt(-3),sqrt(5)) is mu_{6} x <u>. Here mu_n is the group of n-th roots of unity. | |
| May 23, 2010 at 19:59 | comment | added | KConrad | In fact the regulator .96242... for the two quartic fields is just twice the regulator of Q(sqrt(5)): 2*log((1+sqrt(5))/2) = .9624236... | |
| May 23, 2010 at 19:58 | comment | added | KConrad | It's no surprise that those quartic fields have equal regulator. One is the 5th cyclotomic field (the one with discriminant 125) and the other is Q(sqrt(5),sqrt(-3)). Both are quartic CM fields with the same maximal real subfield Q(sqrt(5)). By Prop 4.16 in Washington's Cyclotomic Fields, the regulator of a quartic CM field and the regulator of its maximal real subfield have a ratio that is 1 or 2 (see Theorem 4.12 for the definition of Q that is used in Prop 4.16), and in this case the ratio for both is 2. The fund. unit (1+sqrt(5))/2 in Q(sqrt(5)) is a fund. unit for both quartic fields. | |
| May 6, 2010 at 13:17 | comment | added | danseetea | Yes, I found the fundamental units in Pohst, Weiler, and Zassenhaus. The regulators are indeed equal. That's interesting. | |
| May 6, 2010 at 4:21 | history | answered | Gerry Myerson | CC BY-SA 2.5 |