The quasi-order spectrum of elements with primitive norm, and the exact range of Vega's criterion b² ≠ 2c

Abstract

Let 𝔽_(q) be a finite field and let f=X²+bX+c ∈ 𝔽_(q)[X] be irreducible with c a primitive element of 𝔽_(q). Vega [vega2607] proved that if q+1=4π for an odd prime π, then f is a primitive polynomial if and only if b² ≠ 2c, and asked whether the hypothesis q+1=4π is also necessary. It is not, and the exact list of fields over which the criterion is valid was determined shortly afterwards by Zhu and Wu [zw1]. We give a short treatment of that circle of results in which no polynomial is ever counted: everything about an element θ of a field with qⁿ elements whose norm generates 𝔽_(q)^(×) is controlled by the single integer g(θ)=gcd(ordθ, Q), Q=(qⁿ-1)/(q-1), which for n=2 is the quasi-order of the minimal polynomial of θ. Our new result is the exact spectrum of that invariant: the values realised by the elements outside 𝔽_(q) whose norm generates 𝔽_(q)^(×) are exactly the integers g ≠ 1 admitting a factorisation Q=gm with gcd(m,q-1)=1, for every prime power q, every n ≥ 1 and every g. Both directions come from a generator of the multiplicative group; the realisation half supplies, in complete generality, the existence statement that Zhu and Wu's Remark 7.4 says the proof of their Theorem 7.3 does not provide, and which they obtain instead by counting. The classification of the fields where Vega's criterion holds, a degree-n analogue of Vega's characterisation of the fields where every irreducible with primitive norm is primitive, and negative controls at q=3,5,13,23 all follow from divisor arithmetic. Every statement below is machine-checked in Lean 4 and is quantified over all q; no result rests on a finite case check.

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