The existence of infinitely many cubic fields with class group of exact 2-rank 1

We show that infinitely many cubic fields have class group of 2-rank 1.

💡 Research Summary

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The paper proves that there are infinitely many cubic number fields whose class groups have exact 2‑rank 1, thereby establishing the (p,r) = (2,1) case of the Cohen–Lenstra–Martinet–Malle heuristics for cubic fields. Two independent proofs are given, each relying on recent results concerning the average size and second moment of the 2‑torsion subgroup of class groups in specific families of cubic fields.

The first proof, called the “anomaly approach,” focuses on a thin family of unit‑monogenised cubic fields denoted + B₂₁,₁. These are fields defined by monic irreducible polynomials f(x)=x³+ax²+bx+1 with a,b>0 and (a,b) modulo 4 belonging to {(4,4),(1,2),(2,1)}. Using an effective version of the Birch–Merriman theorem (Bennett) together with an effective Hilbert irreducibility argument from