Assignment:
Let D be a squarefree integer, and let O be the ring of integers in the quadratic field Q(√D). For any positive integer f prove that the set Of = Z[fω] = {a + bfω | a, b ∈ Z} is a subring of O containing the identity. Prove that [O:Of]= f (index as additive abelian groups). Prove conversely that a subring of O containing the identity and having finite index f in O (as additive abelian group) is equal to Of. (The ring Of is called the order of conductor f in the field Q(√D). The ring of integers O is called the maximal order in Q(√D).
Provide complete and step by step solution for the question and show calculations and use formulas.