## 3704 Sheet 6

Dear Sir,

In Problem Sheet 6, for k=Q(sq.root 10)

How did the last bit come about? Why is the class group finally {Ok,p2} Why is p3,p*3 and p5 dropped from the class group? The previous lecturer seemed to have

used mod 5 in his notation of the general element of Ok. Can you please give an answer using the methods from “A few class groups”. In particular can you please

give an answer with the same method as you did for ? Is this possible?

Reply:

I hope you can see that the methods pretty much the same for all the examples. In that problem, there is a list

of ideal of norm less than 7, and every element in the class group is equivalent to one of these. We know that , and hence,

The relation

shows that

then gives

and

Also,

and finally,

So it turns out that

represent all classes in the class group.

One sees thereby that the class group is either trivial (if is principal), or (if is not principal). At this point, one checks that there is no element of with norm 2, using the argument, and hence, that cannot be principal.

### Like this:

Like Loading...

*Related*