Eisenstein Discriminants in Quadratic & Cubic Fields

• Eisenstein discriminants are squarefree integers d ≡ 5 (mod 8) where the real quadratic field’s fundamental unit satisfies ε_d ≡ 1 modulo 2O_K.
• This condition is equivalent to a suite of classical criteria including a Pell-type non-solvability, equality of unit groups, and a tripling in class-number relations.
• The framework connects quadratic fields to real cubic fields via splitting exact sequences, guiding both asymptotic density predictions and computational methods.

Searching arXiv for papers on “Eisenstein discriminants” and closely related usages. In "Quadratic units and cubic fields" (Breuer et al., 9 Jul 2025), an Eisenstein discriminant is a squarefree integer $d\equiv 5 \pmod 8$ such that the fundamental unit $\varepsilon_d$ of the real quadratic field $K=\mathbb{Q}(\sqrt d)$ satisfies $\varepsilon_d \equiv 1 \pmod{2\mathcal O_K}$. This condition belongs to a classical question of Eisenstein and was later sharpened and reframed by Stevenhagen. It links a unit congruence in a real quadratic field to a Pell-type obstruction, to the class groups of the maximal order and the order of conductor $2$, and to the existence and multiplicity of real cubic fields of discriminant $4d$. The same phrase can be misleading outside this context, because other literature uses “Eisenstein” and “discriminant” in different, non-equivalent senses.

1. Definition and ambient quadratic setting

Let

$$K=\mathbb{Q}(\sqrt d), \qquad \mathcal O_K=\mathbb{Z}\!\left[\frac{1+\sqrt d}{2}\right],$$

with $d$ squarefree and $d\equiv 5 \pmod 8$. Because $d\equiv 5 \pmod 8$, the prime $\varepsilon_d$0 is inert in $\varepsilon_d$1. The paper defines an Eisenstein discriminant to be such a $\varepsilon_d$2 for which the fundamental unit $\varepsilon_d$3 satisfies

$\varepsilon_d$4

Equivalently, the fundamental unit lands in the trivial residue class in the quotient $\varepsilon_d$5, which is a cyclic group of order $\varepsilon_d$6 (Breuer et al., 9 Jul 2025).

The associated counting problem is formulated for

$\varepsilon_d$7

Stevenhagen’s conjecture predicts density $\varepsilon_d$8, reflecting the heuristic that the fundamental unit should be equidistributed among the three nonzero residue classes of $\varepsilon_d$9. Since

$K=\mathbb{Q}(\sqrt d)$0

the conjecture says that asymptotically one-third of all $K=\mathbb{Q}(\sqrt d)$1 are Eisenstein (Breuer et al., 9 Jul 2025).

2. Equivalent formulations and class-group structure

For squarefree $K=\mathbb{Q}(\sqrt d)$2, the paper records a package of equivalent classical conditions attributed to Stevenhagen. They show that the defining congruence is simultaneously a unit condition, a Pell-type condition, and a class-number relation.

Condition	Equivalent formulation
Unit congruence	$K=\mathbb{Q}(\sqrt d)$3
Pell-type condition	$K=\mathbb{Q}(\sqrt d)$4 has no solution in odd integers $K=\mathbb{Q}(\sqrt d)$5
Unit-index condition	$K=\mathbb{Q}(\sqrt d)$6 for $K=\mathbb{Q}(\sqrt d)$7
Class-number relation	$K=\mathbb{Q}(\sqrt d)$8

These equivalences identify the order

$K=\mathbb{Q}(\sqrt d)$9

of conductor $\varepsilon_d \equiv 1 \pmod{2\mathcal O_K}$0 as the natural nonmaximal order attached to the problem. The paper further isolates an exact sequence

$\varepsilon_d \equiv 1 \pmod{2\mathcal O_K}$1

where

$\varepsilon_d \equiv 1 \pmod{2\mathcal O_K}$2

Thus, for Eisenstein discriminants, the class group of the nonmaximal order $\varepsilon_d \equiv 1 \pmod{2\mathcal O_K}$3 is exactly three times as large as that of the maximal order. The paper treats the splitting or failure of splitting of this extension as arithmetic data of independent significance (Breuer et al., 9 Jul 2025).

3. Link with real cubic fields

The strongest structural bridge in the paper is the connection with real cubic fields. Stevenhagen’s proposition, quoted there, states that if $\varepsilon_d \equiv 1 \pmod{2\mathcal O_K}$4 and there exists a real cubic field of discriminant $\varepsilon_d \equiv 1 \pmod{2\mathcal O_K}$5, then $\varepsilon_d \equiv 1 \pmod{2\mathcal O_K}$6. Conversely, if $\varepsilon_d \equiv 1 \pmod{2\mathcal O_K}$7, then

$\varepsilon_d \equiv 1 \pmod{2\mathcal O_K}$8

In that case there are

$\varepsilon_d \equiv 1 \pmod{2\mathcal O_K}$9

non-isomorphic cubic fields of discriminant $2$0 (Breuer et al., 9 Jul 2025).

This places Eisenstein discriminants at the interface of quadratic and cubic arithmetic. They are exactly the quadratic discriminants that can support real cubic fields of discriminant $2$1 once the extension splits, and the multiplicity of such cubic fields is governed by the $2$2-torsion of the quadratic class group. The paper therefore compares the counting function for Eisenstein discriminants with

$2$3

where $2$4 is the number of non-isomorphic cubic fields of discriminant $2$5.

The quoted asymptotic for cubic fields is

$2$6

A related asymptotic for the average of $2$7 is

$2$8

These formulas are used in the paper to compare Eisenstein discriminants with cubic fields and to derive improved unconditional upper and lower bounds for $2$9 (Breuer et al., 9 Jul 2025).

4. Counting, conjectural density, and secondary terms

The central asymptotic conjecture is

$4d$0

The numerical computations in the paper extend to $4d$1 and lead to the empirical fit

$4d$2

The data therefore support Stevenhagen’s linear term while exhibiting a noticeable secondary term of size $4d$3 with coefficient about $4d$4 (Breuer et al., 9 Jul 2025).

The comparison with cubic fields is explicit. Defining

$4d$5

the paper obtains

$4d$6

Accordingly, $4d$7 does not look like a centered random walk with fluctuations of square-root type; rather, the observed discrepancy has persistent $4d$8-scale bias. The paper interprets this as evidence that Eisenstein discriminants and cubic-field counts have the same main term but different secondary terms, arising from different mechanisms (Breuer et al., 9 Jul 2025).

The proposed heuristic explanation concerns splitting of the exact sequence above. For $4d$9, if the sequence splits then there are $$K=\mathbb{Q}(\sqrt d), \qquad \mathcal O_K=\mathbb{Z}\!\left[\frac{1+\sqrt d}{2}\right],$$0 cubic fields of discriminant $$K=\mathbb{Q}(\sqrt d), \qquad \mathcal O_K=\mathbb{Z}\!\left[\frac{1+\sqrt d}{2}\right],$$1, and the authors propose the heuristic

$$K=\mathbb{Q}(\sqrt d), \qquad \mathcal O_K=\mathbb{Z}\!\left[\frac{1+\sqrt d}{2}\right],$$2

If this were exact, one might expect $$K=\mathbb{Q}(\sqrt d), \qquad \mathcal O_K=\mathbb{Z}\!\left[\frac{1+\sqrt d}{2}\right],$$3 to behave like an unbiased random walk. The data instead show systematic drift, which the paper interprets as a subtle bias in splitting probabilities, visible even after conditioning on $$K=\mathbb{Q}(\sqrt d), \qquad \mathcal O_K=\mathbb{Z}\!\left[\frac{1+\sqrt d}{2}\right],$$4.

The prime case displays the same pattern. For primes $$K=\mathbb{Q}(\sqrt d), \qquad \mathcal O_K=\mathbb{Z}\!\left[\frac{1+\sqrt d}{2}\right],$$5, the paper finds numerically

$$K=\mathbb{Q}(\sqrt d), \qquad \mathcal O_K=\mathbb{Z}\!\left[\frac{1+\sqrt d}{2}\right],$$6

again suggesting a leading density $$K=\mathbb{Q}(\sqrt d), \qquad \mathcal O_K=\mathbb{Z}\!\left[\frac{1+\sqrt d}{2}\right],$$7 conditional on $$K=\mathbb{Q}(\sqrt d), \qquad \mathcal O_K=\mathbb{Z}\!\left[\frac{1+\sqrt d}{2}\right],$$8, but with a secondary correction of order $$K=\mathbb{Q}(\sqrt d), \qquad \mathcal O_K=\mathbb{Z}\!\left[\frac{1+\sqrt d}{2}\right],$$9 (Breuer et al., 9 Jul 2025).

5. Computational method

The computational problem is to test, for every squarefree $d$0 with $d$1, whether

$d$2

Rather than compute the full fundamental unit, the paper modifies Shanks’s infrastructure method so that all computations are tracked modulo $d$3. The computation proceeds in the reduced-ideal cycle, reducing each generator modulo $d$4 at each step (Breuer et al., 9 Jul 2025).

The crucial simplification is

$d$5

so the entire computation can be encoded in $d$6. The output is $d$7 exactly when $d$8.

The infrastructure procedure uses baby steps and giant steps, but only residue classes of generators modulo $d$9 are required. For the baby-step list, the paper stores tuples $d\equiv 5 \pmod 8$0 and attaches a Bloom filter to the $d\equiv 5 \pmod 8$1-pairs. The Bloom filter quickly rules out most nonmatches and never produces false negatives, which is presented as efficient under GPU memory constraints.

For giant steps, the implementation uses GPU versions of NUCOMP and NUDUPL, together with a selection routine called NUCOMPchoose that decides whether to use NUCOMP, NUDUPL, or direct ideal multiplication. Since small norms can cause overflow or inefficiency in NUCOMP or NUDUPL, the code falls back to direct product when a norm is $d\equiv 5 \pmod 8$2. The implementation is written in C and driven through PyOpenCL, with each GPU kernel handling a separate $d\equiv 5 \pmod 8$3. The paper reports using an Intel Core i5-1235U CPU and Intel Iris Xe GPU, and notes that the code is publicly available on GitHub (Breuer et al., 9 Jul 2025).

6. Terminology, scope, and adjacent meanings

The expression Eisenstein discriminant is not uniform across the literature. In the real-quadratic sense above, it refers to the special family $d\equiv 5 \pmod 8$4 characterized by the congruence $d\equiv 5 \pmod 8$5. By contrast, "Eisenstein integers and equilateral ideal triangles" uses “the discriminant $d\equiv 5 \pmod 8$6” to describe the congruence obstruction attached to the quadratic form

$d\equiv 5 \pmod 8$7

the norm form on $d\equiv 5 \pmod 8$8. In that setting, the relevant statement is that a prime $d\equiv 5 \pmod 8$9 is represented by

$d\equiv 5 \pmod 8$0

if and only if $d\equiv 5 \pmod 8$1 or $d\equiv 5 \pmod 8$2, and the paper explicitly identifies the governing discriminant as $d\equiv 5 \pmod 8$3 (McShane, 2024).

A second possible source of ambiguity is cyclotomic-field terminology. "Perfect Eisenstein integers" computes field discriminants such as

$d\equiv 5 \pmod 8$4

in order to analyze ramification of the distinguished prime $d\equiv 5 \pmod 8$5. That paper states explicitly that it does not develop any special “Eisenstein discriminant” notion for $d\equiv 5 \pmod 8$6 itself (Stumpenhusen, 2022).

These contrasts matter because they separate three distinct uses of closely related language: a unit-congruence condition in real quadratic fields, the discriminant $d\equiv 5 \pmod 8$7 attached to the Eisenstein norm form, and ordinary field discriminants in cyclotomic arithmetic. A common misconception is to identify them. The available papers instead indicate that only the first usage names a specific counting problem of the form studied in (Breuer et al., 9 Jul 2025).

7. Conceptual significance

The arithmetic mechanism emphasized in the modern theory is unusually tight. An Eisenstein discriminant is defined by a congruence condition on a fundamental unit; the same condition is equivalent to a Pell-type non-solvability statement, to equality of unit groups between maximal and nonmaximal orders, and to a factor of $d\equiv 5 \pmod 8$8 in the class-number relation

$d\equiv 5 \pmod 8$9

It is further tied to an exact sequence whose splitting determines whether $\varepsilon_d$00 is the discriminant of a real cubic field, with multiplicity measured by $\varepsilon_d$01 (Breuer et al., 9 Jul 2025).

From the counting perspective, the family appears to have main term

$\varepsilon_d$02

and a secondary term of order $\varepsilon_d$03. This suggests a close but non-identical relationship with the asymptotic theory of real cubic fields. A plausible implication is that Eisenstein discriminants are not merely a subfamily selected by a residue-class heuristic; they also encode finer splitting biases that survive after averaging over the associated cubic-field multiplicities.

In this sense, Eisenstein discriminants form a sharply defined arithmetic family in which quadratic units, nonmaximal orders, class-group extensions, and cubic fields are intertwined. The term is therefore most precise when reserved for the real-quadratic condition introduced above, rather than for the broader and unrelated appearances of “Eisenstein” and “discriminant” elsewhere in arithmetic geometry and the theory of Eisenstein integers.