Oppermann Intervals in F_q[t]: Galois Obstructions, Exact Counts for Small Degree, and the Oppermann Subgroup in Characteristic 2

For a monic f ∈ F_q[t] of degree d, the Legendre interval {f² + s : deg s ≤ d} splits into q Oppermann sub-intervals Of,c = {f² + cf + s : deg s < d}. We ask whether each of them contains an irreducible polynomial. In characteristic 2 the answer is no, and the failures have three Galois-theoretic causes. For d = 1, squares. For d = 2, an AGL₂(F₂) action on the roots of an additive polynomial gives exact counts, which vanish exactly for q = 2 and q = 4^j. For q = 2, d = 4, the Stickelberger–Swan sign of Frobenius. For d ≥ 4 the additive mechanism disappears, because AGL_r(F₂) has no 2^r-cycle when r ≥ 3. For odd q and d = 2, a ternary trace form on Fq⁴ gives π = (q² − 1 − (q − η(−1))η(2e))/4 > 0. In characteristic 2 and d = 3, a count in the tower F_q ⊂ Fq³ ⊂ Fq⁶, together with a cubic Gauss sum, determines every class in closed form. The count is q(q² − 1)/6 for k odd. For k even it is q(q − 1)²/6, q(q² + 1 − 2ε√q)/6 or q(q² + 1 + ε√q)/6, with ε = (−1)k/2. Exact Hayes-FFT counts confirm nonemptiness for odd q (up to q = 3, d ≤ 16) and find no further empty sub-intervals in characteristic 2 (up to q = 2, d ≤ 26). The Bank–Bary-Soroker–Rosenzweig theorem then gives nonemptiness for every d ≥ 3 once q is large. In characteristic 2 the Oppermann classes form a subgroup H_d with G_d/H_d ≅ F_q⌊d/2⌋, cut out by the odd power sums. Its annihilator yields Artin–Schreier sums. The mean of Ψ over H_d drifts exactly by −2q² + 2q (d = 4, k odd) and by q² − q (d = 5). Its variance is a coherent sum over annihilator cosets, maximal at d = 2. Code and data: https://github.com/Ruqing1963/oppermann-function-fields