Artin’s primitive-root conjecture for quadratic number fields

Let $K$ be a quadratic number field and let $\alpha\in K^\times$ be neither a root of unity nor a square in $K$ . Then the set $\{\mathfrak p: \mathfrak p\nmid(\alpha),\ \langle\overline{\alpha}\rangle=(\mathcal O_K/\mathfrak p)^\times\}$ of prime ideals for which the reduction $\overline{\alpha}$ is a primitive root modulo $\mathfrak p$ is conjectured to have a positive natural density. Equivalently, $\operatorname{dens}(\alpha)>0$ , where $\operatorname{dens}(\alpha)$ denotes this density.

References

Primary source

Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg

Additional references

Uniform bounds for the density in Artin’s conjecture for quadratic number fields — Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg — Chi Wa Chan

analytic-number-theory number-theory

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed progress

A new paper reports stronger frequency bounds, but the quadratic-field conjecture has not been unconditionally proved.

Artin posed the primitive-root conjecture in 1927; its quadratic-field form asks for density information on primes where an element has maximal multiplicative order. The unconditional conjecture remains open.

Known results

Roskam (2002): under GRH, density exists for the relevant primes in a quadratic field, with necessary and sufficient conditions for positivity.
Perucca and Shparlinski: under GRH, density bounds depending only on the number field are obtained; over $\mathbb{Q}$ , explicit optimal bounds are given.
Hooley (1967): conditional proof of the classical integer case under suitable GRH hypotheses.

2026 journal paper

Chi Wa Chan’s paper, Uniform bounds for the density in Artin’s conjecture for quadratic number fields, reports new uniform bounds for the quadratic-field density. The DOI record gives no abstract or numerical statement, so this is claimed progress rather than a verifiable account of the bound.

Current status (as of August 2026): Conditional density results are known under GRH, and a 2026 paper claims new uniform bounds, but no unconditional solution of the quadratic-field conjecture is recorded.

Sources

No solutions have been posted yet.