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Behavior of N_n(F_k) along towers of quadratic finite field extensions

Characterize the behavior (e.g., monotonicity, growth, stabilization) of the sequence (N_n(F_k))_k when n is fixed and F_{k+1} is a quadratic extension of F_k for each k in a tower of finite fields F_1 ⊂ F_2 ⊂ ⋯.

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Background

The authors introduce N_n(F) and show that it depends sensitively on the ground field. In particular, they observe finiteness for finite fields and infiniteness for certain infinite fields.

To better understand how N_n(F) varies with the field, they pose the question of the asymptotic behavior of N_n(F_k) along towers of finite fields obtained by successive quadratic extensions.

References

This facts motivated us to open the following new questions, that we expect to answer in future works:

Q4: If we left n fixed and we take a tower of finite fields _1\subset _2\subset \dots, where _{k+1} is a quadratic extension of _k for each k. How is the behaviour of the sequence of numbers (N_n( _k))_k?

— Invariants for isomorphism classes in the category $\bcalNT$ (2508.00084 - Maturana, 31 Jul 2025) in Introduction (Section 1)