I strongly suspect this question has a very straightforward answer.
Let $M = \mathbb{Q}(\sqrt{a_1},\dots,\sqrt{a_k})$ be a large multiquadratic field. In this setup we assume it is infeasible to compute $M$ directly, because it has degree $2^k$ over $\mathbb{Q}$.
Let $\mathfrak{i}_1 = (\alpha), \mathfrak{i}_2 = (\beta)$ be principal ideals of $M$ and $\alpha,\beta$ proper elements of $M$ with sparse support. By this I mean $\alpha,\beta$ are not in a subfield of $M$ and the number of monomials in $\alpha,\beta$ is polynomial in $k$.
Let $\mathfrak{i}_3 = \mathfrak{i}_1 + \mathfrak{i}_2$. Then $\mathfrak{i}_3$ has a two-element form $\mathfrak{i}_3 = (\gamma, \delta)$, where $\gamma \in \mathbb{Z}$ and $\delta \in M$.
Apparently if we take the norm of $\mathfrak{i}_3$ to the next subfield down, it is an ideal of (the ring of integers of) this lower field we will call $M_{-1}$, and the two-element form can be much simpler. However I only care about the integer norm part, and my question is whether/how it can be computed efficiently, i.e. in time polynomial in $k$.
Here is an example in Sage:
r = [8158, 305, 490, 677, 866] M = QQ gens = [] names = [] for i, val in enumerate(r, start=1): gen_name = "a"+str(i) M = M.extension(x^2 - val, names=gen_name) gens.append(M.gen()) names.append(gen_name) globals().update(dict(zip(names, gens))) alpha = a1 + a2 + a3 + a4 + a5 + 5 beta = a1 + a2 - a3 - a4 + a5 + 7 i1 = M.ideal(alpha) i2 = M.ideal(beta) i3 = i1 + i2 i3.relative_norm() which constructs $M = \mathbb{Q}(\sqrt{8158}, \sqrt{305}, \sqrt{490}, \sqrt{677}, \sqrt{866})$ and prints
Fractional ideal (41, a4 - a3 + a2 - a1 - 22) showing the simple two-element form of $\mathcal{N}_{M/M_{-1}}\left(\mathfrak{i}_3\right)$.
So my question is how can we efficiently compute the ideal sum $\mathfrak{i}_3 = \mathfrak{i}_1 + \mathfrak{i}_2$, and take the norm down to $M_{-1}$. I am not even asking for its full representation, only the integer norm part and I suspect it is very efficient to compute.
By contrast computing absolute norms down to $\mathbb{Q}$ of elements in $M$ is unfeasible in general because of coefficient explosion, each time we descend to a subfield the number of monomials doubles.
Cohen's sequel book Advanced Topics in Computational Number Theory has several algorithms for relative number fields but I can't pick out this exact case and it may be simpler than the general case. However the relative HNF (Hermite Normal Form) for ideals may be especially relevant. I would pick apart the source code for Sage or pari-gp because I think it will have an efficient method in this case but decided to ask here instead first.
Edit: It seems to be enough to compute the ideal sum
$$(\alpha\sigma(\alpha)) + (\beta\sigma(\beta))$$
where $\sigma$ is the automorphism of $M$ fixing $M_{-1}$ (in the example sending $\sqrt{866} \mapsto -\sqrt{866}$). This is now a very clear task: Can the norm of this ideal sum be computed efficiently? It would be nice for someone to finish it out (I will try).
