I started reading Class group after some one's advice ,so I got the following doubts,I would be happy if someone clarify the doubts,

I understood that the class group measures the failure of the Unique factorization ,here are the doubts I have got,

  1. What was the use of bothering(I refer to inventing) about the Unique factorization,even though i know that it was invented in the course of proving the Fermat's Last theorem by Kummer,as the Fermats equation can be splitted into the Cyclotomic fields I mean $\mathbb{Z[\omega]}$,

    So are there any other applications of the ideal class group???,I mean what are the other areas that ideal class group has got its application???

  2. And moreover we know that the Tate-Shafarevich group is the analogue of the class group of the number field ,but when i read the Definition of the Tate-Shafarevich Group $Ш(E/K)=\mathrm{Ker}(H^1(K,E)\mapsto \prod_{v}H^1(K_v,E))$,the non-trivial elements of the Tate-Shafarevich group can be thought of as the homogeneous spaces of $A$(where $A$ is an Abelian Variety defined over $K$) that have $K_v$-rational points for every place $v$ of $K$, but no $K$-rational point. And in terms of Homogenous Spaces it is defined as $Ш(E/\mathbb{Q})=Sel(E/\mathbb{Q})/(E(\mathbb{Q})/2E(\mathbb{Q}))$, my main question is that "The Class group on one hand measures the failure of Unique factorization ,and the Tate-Sha group on the other hand measures the extent failure of local global principle ,i mean set of homogeneous spaces with local points but no global point",

    How can one relate the Tate-Sha group that measures failure of Hasse principle to the class group that measures the failure of unique Factorization ???,to be precise,how can the Tate-sha Group and Class group can be thought as analogues??,can anyone explain the reason behind it clearly

thanks a lot,

note:Please,i have been having 2 negative votes without any comment ,

anyone who gives downvote are requested to post comments in order to rectify myself

  • $\begingroup$ it was the advice by mr.George,and i am thankful to him $\endgroup$ – IDOK Sep 16 '11 at 15:43
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    $\begingroup$ i clearly mentioned in comments that give the reason for downvotes,please do it $\endgroup$ – IDOK Sep 16 '11 at 16:50
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    $\begingroup$ Now, unique factorization was not invented for the purposes of Fermat's last theorem. It is a fact proved millenniums ago that any integer has a unique factorization into prime power factors. I had earlier requested you to please read a proof of this first. Kummer wanted an analogous fact; but as others explained to you, it breaks down for algebraic integers. And then the ideals etc were invented. But first please read a proof of unique factorization of integers. I frankly do not understand the point of someone who is ignorant of that theorem doing number theory. $\endgroup$ – George Sep 16 '11 at 17:02
  • $\begingroup$ @George:sir how can you think that i asked without reading about it sir,i followed your advice,and what i said was right ,but its onyl due to the result of the sense in which you took it,Unique factorization Property surely played a role a major role in Fermats last theorem,Kummer actually Tried to prove the Fermats last theorem by considering whether the unique factorization of $\mathbb{Z}$ and $\mathbb{Z[i]}$ generalizes to the ring $\mathbb{Z}[\omega]$ ,ok sir?? $\endgroup$ – IDOK Sep 16 '11 at 17:14
  • $\begingroup$ i have read that proof sir,and understood it,i dont know what made you understand that i didnt read that proof,@george $\endgroup$ – IDOK Sep 16 '11 at 17:24

I have found your questions interesting despite that they could be phrased better and reflect more familiarity with the topic of the question. I have recently seen, althought this is not news, that the class group can be used to factorize integers: http://cr.yp.to/bib/1982/schoof.html , and it is an interesting group in its own right. What does the Tate-Shafarevich group for elliptic curves have to do with the ideal class group? It has been shown that if we try to use a particular method of finding a rational point of an elliptic curve known as descent, and if we try to find integer points of a Pell conic by an analogous method, the methods being find a point by solving a related curve called a descendant, this doesn't always give us a point. In other words, the descent is obstructed. The nature of the obstruction is a group called the Tate-Shafarevich group. It has been shown that for Pell conics, the Tate-Shafarevich group is isomorphic to a particular subgroup of the ideal class group. In the case of elliptic curves I don't think that it is known whether there is an analogue of the ideal class group for which the Tate-Shafarevich group may be identified.


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