Let be a squarefree integer, and let be the ring of integers in the quadratic field . (I.e. .) For any positive integer prove that the set is a subring of containing the identity. Prove that . (The index as an abelian group.) Prove conversely that a subring of containing the identity and having (as a subgroup) finite index is equal to . (The ring is called the order of conductor in the field . The ring of integers is called the maximal order in .)
First we show that is a subgroup of . To see this, note that . Now let and be in . Then , so that by the subgroup criterion, is an additive subgroup of .
Now it is easy to see that or , depending on whether is not or is 1 mod 4. Moreover, if is 1 mod 4, then is an integer. Thus we have or , depending on whether is not or is 1 mod 4. In either case, . Thus is a subring.
Now we will show that . Define a mapping by . We claim that is an additive group homomorphism. To see this, note that . Moreover, is surjective since for all , . Consider now , where denotes the natural projection. We claim that , as we show. If mod , then for some integer . Thus . Clearly mod . By the First Isomorphism Theorem for groups, then, , hence .
Suppose now that is a subring containing 1 and having finite index . If , then trivially, . If , then since , , and in particular, for all integers .
Let . Since has index , for all . (Here, denotes the -fold sum of .) Specifically, ; thus for all , and we have .
Now . Thus , and we have .