Monday, 13 October 2008

complex multiplication - A problem of Shimura and its relation to class field theory

I would like to write some kind of summary of the above two answers. There is nothing new here.



Consider mathbbQsubsetmathbbQ(zeta7),(and we fix zeta7=efrac2pii7) this is an abelian Galois extension, and the Galois group is (Z/7Z)times. Frobenius element over p for pneq7 (7 is the ramification) acts on zeta7 by sending it to its p-th power.



Now consider alpha=zeta7+zeta17, which is in mathbbQ(zeta7), consider all its Galois conjugates, which are precisely alpha2=zeta2+zeta2, alpha3=zeta3+zeta3. So we have mathbbQsubsetmathbbQ(alpha) is Galois.



Now Frobp maps alpha to zetap+zetap, which is equal to alpha if and only if pequiv1,1(mod7). And they are precisely those primes that are totally split in mathbbQ(alpha). For those primes, Z/p=mathbbOmathbbQ(alpha)/p, thus we can always find some n, s.t., alphaiequivn(modp), which is equivalent to say that F(x) totally split over Z/p=Fp.



I am glad to know this problem and answer since I finally found an explicit example of cyclic Galois extension of mathbbQ...(which I had been wondering for a while...)



We can also do the similar things for p=13. just take beta=theta+theta5+theta8+theta12,where theta is the 13-th root of unity, then Q(beta) is again a cyclic Galois extension of Q.




when I said cyclic above, I meant for cyclic of order 3. Thank Peter for pointing out.

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