PRINCIPAL — where a quadratic field's first principal prime sits

The object: a quadratic field K = Q(√D), D a fundamental discriminant, and its split primes whose places are principal. This page states how far down the least of them can sit, which the sign of D decides, and how a finite cut's split primes depart from their classes' even shares. Its engine is principal.py.

An odd prime p with χ_D(p) = +1, χ_D the Kronecker symbol (D/·), splits as P·P̄, and P is principal when P = (α); over a real field that is principal as an ideal (the wide sense) unless the narrow group is named. L₁(D) is the least odd prime with χ_D(p) = +1 and P principal; its P is a principal rank-1 place of least odd norm, and at any principal rank-1 place a greedy walk whose moves must be principal keeps the carrier ladder, climbed through carriers (rank-1 and carrier as CASCADE's lead defines them, the claim at CASCADE: The escape is a conjunction). The principal form of discriminant D is x² + b₀xy + c₀y², b₀ = D mod 2, c₀ = (b₀ − D)/4, and N(x + yω) is its value at (x, y), ω = (b₀ + √D)/2. So P is principal exactly when p or −p is a value of that form, that is when

4p  =  |u² − D·y²|,      u = 2x + b₀y.

The answer, in two lines: over an imaginary field the form is definite, so a principal prime is at least |D|/4 and, to |D| ≤ 20000, the median least one sits inside the floor's range over its band of |D| (an observation), while over a real field the form is indefinite and there is no floor at all; and over a population of fields the degree-1 primes at a finite cut stand above their even shares on the classes that are neither squares nor 2-torsion, by the weight of the prime-ideal powers the count leaves out, which lands almost all on the other classes (a pattern): counting those powers removes 93–104% of that excess over imaginary fields, where at 10⁴ every (class number, cell) reading with 30 fields or more then lands within 0.03 of 1, and 84–97% over real ones in the narrow class group. What still departs over real fields sits, at the low cuts, mostly on the narrow trivial class and on the class of principal ideals whose generators all have negative norm, the latter still the largest departure at 10⁴, and is open.

The floor

Tier: theorem.
Verifier: proof; principal.py::section_floor.

Over an imaginary field, L₁ ≥ |D|/4, and L₁ ≥ (|D| + 1)/4 at odd D. The floor is met, L₁ = (|D| + 1)/4, exactly when D is odd and (|D| + 1)/4 is an odd prime; at even D it is never met.

Proof. Let P over p be principal, so 4p = u² + |D|y². If y = 0 then 4p = u² and p is a square, which it is not; so y ≠ 0 and 4p ≥ |D|. At odd D, u ≡ b₀y ≡ y mod 2, so a y of either parity gives 4p ≥ |D| + 1. For equality at odd D, the form's value at (0, 1) is c₀ = (|D| + 1)/4, and D = 1 − 4c₀ makes c₀ prime to D: an odd prime c₀ is unramified and represented, hence split and principal, and nothing lower can be. At D ≡ 0 mod 4, 4p = |D| forces u = 0 and |y| = 1, so p = |D|/4, which divides D and ramifies. Conversely L₁ is an odd prime, so it meets (|D| + 1)/4 only where that value is one.

The floor's proof uses the size of the norm form and never the class group. Its corollary is that the imaginary fields with L₁ ≤ B form a finite set, since none with |D| > 4B has L₁ ≤ B. Over all 6079 imaginary fields with |D| ≤ 20000 the scan meets the floor at 499, each at the criterion; the median L₁ runs 181, 587, 1163, 2269, 4483 over the |D| bands (0, 1000], (1000, 2500], …, (10000, 20000], each below a quarter of its band's top, so inside the floor's range over the band; by the floor a scan for a principal prime finds none below |D|/4.

No floor over a real field

Tier: theorem; observation (the readings).
Verifier: proof; principal.py::section_family, principal.py::section_real.

Over a real field no lower bound on L₁ growing with D exists: L₁ = 3 at every squarefree D > 0 of the form m² − 12 or m² + 12 with m odd and 3 ∤ m, and there are infinitely many.

Proof. With m odd, m² ≡ 1 mod 8, so D ≡ 5 mod 8 and b₀ = 1. Put x = (m − 1)/2: then N(x + ω) = x² + x − (D − 1)/4 = (m² − D)/4 = ±3. D ≡ m² mod 3 is a nonzero square, so χ_D(3) = +1, and x + ω generates a place over 3. So L₁ = 3 whenever D is fundamental, that is squarefree. Writing m = 6n ± 1, D is an irreducible quadratic in n with no fixed square divisor: it is odd, prime to 3, and modulo ℓ² for ℓ ≥ 5 a quadratic with unit leading coefficient 36 does not vanish at every n. Such a quadratic takes squarefree values with positive density (Ricci, Rend. Circ. Mat. Palermo 57, 1933; the case z² + k is Estermann's, Math. Ann. 105, 1931).

The difference between the two signs is the sign in 4p = |u² − Dy²|: a definite form's values grow with |D|, an indefinite one takes small values far out. Over a quadratic field that is the unit rank, since a finite unit group and a definite norm form are both the condition D < 0; higher up the two come apart (a totally imaginary quartic field is definite with unit rank 1), and which of them a floor needs there is left open. The class engine finds a wide-principal place over 3 at each of the 627 members below 10⁶. Over all 4865 real fields with D ≤ 16000 every one has L₁ < 10⁴, and the median runs 7, 11, 11, 11, 11 over the D bands (0, 1000], (1000, 2500], …, (10000, 16000]: from the second band on it stays at 11. A complex cubic field has unit rank 1 and an indefinite norm form, and over all 888 whose discriminant is at most 6000 in absolute value a principal degree-1 place lies over an odd unramified prime below 250, the largest such least prime 131; to 50000 it is 823, both over class groups certified numerically, not proved (TRIPLE; triple.py::section_controls, triple.py::section_coverage).

Where the omitted prime powers land

Tier: property.
Verifier: proof; principal.py::section_imag.

Chebotarev in the Hilbert class field H/K, whose group is the class group Cl of order h, speaks of the count Π_C(x) of all prime-ideal powers P^k of K with N(P^k) < x in the class C, each weighted 1/k: its main term is Li(x)/h for every class. The degree-1 count read on this page is of the split places alone, so it omits three families, a ramified prime among them though its degree is 1, and each lands on known classes:

The remaining powers, each weighted 1/k at norm below x, land as follows: even powers of a split place on squares, powers of inert and ramified primes on the trivial class and Cl[2], and only the odd powers k ≥ 3 of split places, p < x^(1/k) (the cubes p < 22 at x = 10⁴), anywhere. So a class in neither 2Cl nor Cl[2] loses only that sliver of odd-power weight, while a class in 2Cl or Cl[2] loses besides the weight of every family and power above that lands on it. Split places below x number about π(x), so a class holds about π(x)/h of them, and the trivial class loses about π(√x)/2 inert primes plus, were the split places below √x spread evenly over the classes, the share |Cl[2]|/h of the split squares' weight π(√x)/2: to leading order under that spread, a deficit against its Chebotarev share Li(x)/h of

(h + |Cl[2]|)/2 · π(√x)/π(x),

affine in h at a fixed |Cl[2]|, the ramified primes' weight on Cl[2], 1/k for each power R^k of norm below x, left out. Read against the field's realised total over h, itself short of Li(x)/h by about π(√x)/h, the trivial class falls short by that deficit less π(√x)/π(x), to the same order. Over a real field the same holds in the narrow group Cl⁺, of order h⁺, since q and r are generators of positive norm.

The shortfall is the prime powers

Tier: pattern (imaginary fields |D| ≤ 4000 with h ≥ 2, the (class number, cell) readings of 30 fields or more at h = 4, 6, …, 16, 20, 24, 28; real fields D ≤ 16000 with h⁺ ≥ 2, in the narrow group; the cuts 250, 400, 630, 1000, 2500, 10⁴).
Verifier: principal.py::section_imag; principal.py::section_real; principal.py::section_race.

Per field the level of a set of classes is its mean count per class over the field's total divided by h. The cells are T, the trivial class; A, the other classes of Cl[2], over a real field less one read alone below; S, the squares outside Cl[2]; G, the classes in neither 2Cl nor Cl[2], which lose only the sliver of odd-power weight. Read over a population of fields, where the explicit formula's oscillating remainders average out and its prime-power term does not (the floor, one-signed, is read apart below, and the zeros at s = 0 under Open fronts set one more term), the degree-1 levels depart from 1 by what the omitted weight predicts, to the percentages below. Counting Π_C instead removes 93% to 104% of the G cell's excess at every cut over imaginary fields and 84% to 97% over real ones; over imaginary fields at 10⁴ every (class number, cell) reading with 30 fields or more lands in 0.97 to 1.02, and over real fields a remainder sits mostly on two classes at the low cuts. A field's levels average to 1 over its classes, so a cell is read against its siblings, and which classes make up each cell is set by the group's shape, not its order: a stratum named by the class number alone mixes shapes. At a prime h a field has two cells, so the non-trivial one is fixed by the trivial one, at level (h − η)/(h − 1) with η the level of T, and casts no vote of its own: a fit across strata that counts it as one is testing the trivial class twice.

Over the 1208 imaginary fields with h ≥ 2 the G cell, present at 880 of them, reads, raw against counted in prime powers,

cut        250     400     630     1000    2500    10⁴
raw        1.186   1.145   1.106   1.089   1.057   1.023
Π          1.013   1.002   0.996   1.000   1.003   0.999

at 42 standard errors or more above 1 raw, with 93% to 104% of the excess removed. The raw excess falls by a factor of 8.0 from cut 250 to 10⁴; in the same level units the term's leading piece π(√x)/π(x) falls by 5.6, the gap to 8.0 not printed apart, where a fixed count of primes would fall by π(10⁴)/π(250) = 23 and a constant level by 1. At 10⁴ all 39 (class number, cell) readings with 30 fields or more read 0.97 to 1.02 counted in prime powers, while the raw trivial class runs from 0.95 at h = 4 down to 0.75 at h = 28. Averaged over all 1208 fields at 10⁴, its raw deficit 1 − η is 0.208; the section above's deficit less π(√x)/π(x), the realised total's own shortfall, averages 0.204 field by field, a ratio of 1.02, a comparison settled after the run (the deficit as first predicted, 0.2245, reads 0.93). Neither counts the floor, which empties the trivial class of split places below |D|/4 ≤ 1000, and why that loss does not show in the ratio is not derived. Below x ≈ |D|/4 the trivial class holds no split place, so η at small cuts is read, never predicted.

Over the 4107 real fields with h⁺ ≥ 2, the G cell present at 1069 of them, the same count removes 84% of the G excess at cut 250 and 97% at 10⁴ (1.0255 raw, 1.0009 counted). What it leaves is not spread evenly: counted in prime powers, the trivial class reads 1.07 at cut 250 and 1.005 at 10⁴, and k₀, the narrow class of a principal ideal whose generators all have negative norm, 0.82 and 0.99; the other cells lie within 0.035 of 1 at cut 250 and 0.005 at 10⁴. That remainder is open (Open fronts). The classical instance is the control: over the 77 odd prime moduli below 400 at x = 10⁶ the non-residue lead, the count of primes below x that are non-residues less the count that are residues, over the two counts' mean, reads 0.00180 ± 0.00031 (a standard error) against π(√x)/π(x) = 0.00214, and −0.00042 ± 0.00031 counted in prime powers.

In the character basis, a weight spread evenly over 2Cl, as the split squares' weight is taken to be above, transforms onto the characters trivial on 2Cl, the genus characters, since the indicator of a subgroup transforms onto the characters trivial on it.

The bias against principal primes

Tier: known.
Source: M. Aoki and S. Koyama, Chebyshev's bias against splitting and principal primes in global fields, J. Number Theory 245 (2023), 233–262, Corollary 3.5 and Example 3.7.

Assume DRH(A), the deep Riemann hypothesis in Aoki and Koyama's form (A): the convergence of the Euler products of Gal(H/K)'s L-functions at s = ½, none vanishing there. Then over the prime ideals P of K with N(P) ≤ x,

Σ_{P non-principal} 1/√N(P)  −  (h − 1)·Σ_{P principal} 1/√N(P)
    =  (|Cl/2Cl| − 1)/2 · log(log x)  +  c  +  o(1),

a bias against principal primes whenever h is even; over an imaginary field |Cl/2Cl| = 2^(t − 1), t the number of primes dividing D (genus theory). Aoki and Koyama's race is weighted and asymptotic; the unweighted count of degree-1 primes at a finite cut, class by class, is the accounting above.

Open fronts

What the prime-power count leaves over real fields sits mostly on the trivial class and on k₀, and k₀ is the class the archimedean place names. It exists when the fundamental unit has norm +1, so that h⁺ = 2h, and then complex conjugation, which fixes the real field K, moves its narrow class field H⁺: its image in Gal(H⁺/K) = Cl⁺ is k₀. The two real places give the same image, since their classes differ by that of (α) for a totally negative α, and (α) = (−α) has the totally positive generator −α. So a character χ of Cl⁺ with χ(k₀) = −1 is odd at both real places: its L-function carries the gamma factor Γ_R(s + 1)² and does not vanish at s = 0. One with χ(k₀) = +1 carries Γ_R(s)², and a double zero at s = 0 if it is nontrivial; ζ_K has a simple one. Let ψ(x, χ) be the sum of χ(P^j)·log N(P) over the prime ideal powers P^j of norm at most x. In the explicit formula a zero of order m at s = 0 puts −m·log x into ψ(x, χ), and averaging over the characters against χ̄(C) gives −(1 − 1/h⁺)·log x at C = 1 and at C = k₀ and +(1/h⁺)·log x at every other class. The archimedean place alone sets the two classes 1 and k₀ one log x below the rest in ψ; when the fundamental unit has norm −1 there is no k₀, every character is even, and the term is −(2 − 1/h)·log x on the trivial class alone, two log x below the rest. Over an imaginary field every nontrivial class character carries Γ_C(s) and vanishes once at s = 0, so the same term sets the trivial class one log x below the rest there, where pooled over the 1208 fields it reads 0.987 counted in prime powers at 10⁴, a reading that also carries the floor's loss. Whether that term is the whole of k₀'s deficit at the cuts read, what it becomes in a count of primes rather than in ψ, and why over real fields the trivial class reads high where k₀ reads low, is not derived. Nothing further is claimed of what the prime-power count leaves.