Let $O_L$ and $O_K$ be Dedekind domains and Let $L$ and $K$ denote their corresponding field of fractions. Further, let $ L/K $ be a finite separable extension.
For a fractional ideal $J$ of $O_L$, we define its ideal norm, $N(J) = [O_L : J ] _R $ using the concept of module index. (Refer Algebraic number theory by Cassels and Frolich).
I was also able to prove that the above definition of $N(J)$ is equivalent to the one defined as $N(J) = \sum_{a \in J} N_{L/K}(a) $, where $N_{L/K}(a)$ is the field norm.
My claim: For any fractional ideal I of $O_K$, $ \ N(I \cdot O_L) = I^n $, where $ n = [L:K].$ I am able to prove this claim using the first definition (-the module index one)
However, when $L/K$ is a degree 2 galois extension, and I is not principal, say, $ I = (a,b)$, as $I^2 = (a^2, b^2, ab)$, I am unable to come up with an element $\alpha \in I$ such that $N_{L/K}(\alpha) = ab $.
I feel that such an $\alpha$ must exist using the equivalent definitions of the Ideal norm.
