Construction and classification of \(p\)-ring class fields modulo \(p\)-admissible conductors

1. Introduction

The aim of this article is to present an entirely new technique for the construction and classification of non-Galois fields \(L\) of odd prime degree \(p\) as subfields \(L< K_f\) of a \(p\)-ring class field \(K_f\) modulo a \(p\)-admissible conductor \(f\) over a quadratic base field \(K\). The innovative idea underlying this new method is the fact that, if the Galois closure \(N\) of such a field \(L\) is absolutely dihedral of degree \(2p\) with automorphism group \(\mathrm{Gal}(N/\mathbb{Q})\simeq D_p=\langle\sigma,\tau\mid\sigma^p=\tau^2=1,\tau\sigma=\sigma^{-1}\tau\rangle\), then \(N\) is relatively cyclic of degree \(p\) with group \(G=\mathrm{Gal}(N/K)\simeq C_p=\langle\sigma\rangle\) over its unique quadratic subfield \(K=\mathrm{Fix}(\sigma)\) and can be viewed as an abelian extension modulo some conductor \(f\) over \(K\) within the scope of class field theory [1,2,3,4].

The construction process for the fields \(L\) is implemented as a program script for the computational algebra system Magma [5,6,7] using the class field theoretic routines by Fieker [3], and the normal fields \(N/L\) are classified according to the cohomology \(\hat{H}^0(G,U_N)\) and \(H^1(G,U_N)\) of their unit group \(U_N\) as a Galois module over \(G\) [8,9,10].

For \(p\ge 5\), the results are completely new, whereas for \(p=3\), they admit an independent verification and a class field theoretic illumination of classical tables of cubic fields by Angell 1972 [11,12] and 1975 [13,14], Ennola and Turunen 1983 [15,16], Llorente and Quer 1988 [17], Fung and Williams 1990 [18,19], and Belabas 1997 [20]. However, in contrast to these well-known tables, where the focus was on the computation of fundamental systems of units and the structure of ideal class groups [11,12,13,14,15,16,18], or even only of generating polynomials and prime decompositions [17,20], our innovative database establishes an arrangement according to conductors with an increasing number of prime factors, pays attention to the phenomenon of multiplicities of discriminants [21,22,23,24,25], and constitutes the first classification into \(9\), respectively \(3\), differential principal factorization types of totally real, respectively simply real, cubic number fields [8,9,10,26,27]. This is a progressive new kind of structural information which has never been provided for algebraic number fields before, except for pure cubic fields [28,29,30,31] and pure quintic fields [8], but the present paper emphasizes the advanced viewpoint of classifying an entire ring class field \(K_f\) by its associated heterogeneous multiplet \(\mathbf{M}(K_f)\) of dihedral fields with various conductors \(c\mid f\).

2. Heterogeneous multiplets of objects and invariants

Let \(K=\mathbb{Q}(\sqrt{d})\) be a quadratic base field with positive or negative fundamental discriminant \(d=d_K\equiv 0,1\,(\mathrm{mod}\,4)\), essentially squarefree except possibly for the \(2\)-contribution \(v_2(d)\). Suppose that \(p\) is an odd prime number and \(f\ge 1\) is a \(p\)-admissible conductor over \(K\) [21,25]. Then the \(p\)-ring class field \(K_{p,f}\) mod \(f\) of \(K\) contains all cyclic relative extensions \(N/K\) with some conductor \(c\mid f\) which are absolutely dihedral with automorphism group \(\mathrm{Gal}(N/\mathbb{Q})\simeq D_p\) over the rational number field \(\mathbb{Q}\). The crucial concept underlying this entire paper is the collection of all these dihedral fields in a heterogeneous multiplet \(\mathbf{M}(K_{p,f})=\lbrack(N_{c,i})_{1\le i\le m_p(K,c)}\rbrack_{c\mid f}\) according to the \(p\)-multiplicities \(m_p(K,c)\) [21,25], which satisfy the relation \(\sum_{c\mid f}\,m_p(K,c)=\frac{p^{\varrho_{p,f}}-1}{p-1}\) in terms of the \(p\)-ring class rank \(\varrho_{p,f}\) modulo \(f\) of \(K\). Since our principal aim is the classification of \(p\)-ring class fields \(K_{p,f}\), it is essential to distinguish between a multiplet of objects (expressing the multiplicity of the discriminants \(d_N\)) and a corresponding multiplet of invariants (expressing the Galois cohomology of the unit groups \(U_N\) and differential principal factorizations of the fields \(N\)).

Definition 1. By the type of the \(p\)-ring class field \(K_{p,f}\) modulo \(f\) of \(K\) we understand the pair \((\mathrm{Obj}(K_{p,f}),\mathrm{Inv}(K_{p,f}))\) of heterogeneous multiplets

consisting of all absolutely dihedral fields \(N_{c,i}\) with conductors \(c\) dividing \(f\) as objects and their differential principal factorization types (DPF types) \(\tau(N_{c,i})\) as invariants [8,9].

3. Homogeneous multiplets of unramified extensions

The unique situation where the heterogeneous multiplets degenerate to homogeneous multiplets occurs for unramified relative extensions \(N/K\) with conductor \(f=1\) which has only itself as a divisor \(c\mid f\). In this unramified case, which implies positive \(p\)-class rank \(\varrho_p=\varrho_{p,1}\ge 1\) of the quadratic base field \(K\), there occur at most two possible differential principal factorization types.

Theorem 1. An unramified cyclic extension \(N\) with odd prime degree \(p\) of \(K\) possesses the conductor \(f=1\) without any prime divisors. For a totally real field \(N\), there are two cases:

• 1. If the \(p\)-class rank of \(K\) is \(\varrho_p=1\), then \(N\) is of type \(\delta_1\).
• 2. If the \(p\)-class rank of \(K\) is \(\varrho_p\ge 2\), then two types \(\alpha_1\) and \(\delta_1\) are possible for \(N\).

If \(N\) is totally complex, then \(N\) is of type \(\alpha_1\), independently of the \(p\)-class rank of \(K\).

Proof. Since the conductor \(f=q_1\cdots q_t\) is essentially the square free product of all prime numbers \(q_i\in\mathbb{P}\), whose overlying prime ideals \(\mathfrak{q}_i\in\mathbb{P}_K\) are ramified in \(N\), the following chain of equivalent statements is true: \(N/K\) is unramified \(\Longleftrightarrow\) None of the prime ideals of \(K\) ramifies in \(N\) \(\Longleftrightarrow\) The conductor \(f=1\) has no prime divisors, i.e., \(t=0\).

Now we use the fundamental equation in [9,Corollary 5.1] and the estimates in [9, Corollary 5.2] for the decision about possible types of principal factorizations. If \(f=1\), then there neither exist absolute principal factorizations in \(L/\mathbb{Q}\), since \(0\le A\le\min(t,2)=0\), nor relative principal factorizations in \(N/K\), since \(0\le R\le\min(s,2)=0\), where \(s\le t\) denotes the number of prime divisors \(q_i\) of \(f\) which split in \(K\). Consequently, the fundamental equation degenerates to \(U+1=C\) with \(1\le U+1\le 2\), which implies \(1\le C\le\min(\varrho_p,2)\). Thus, only type \(\delta_1\) with \(C=1\) is possible for \(\varrho_p=1\), whereas type \(\alpha_1\) with \(C=2\) can arise additionally for \(\varrho_p\ge 2\).

4. Conductors with a single prime divisor

For a regular prime conductor \(f\), only two cases are possible.

Theorem 2. Let \(K\) be a quadratic base field with \(p\)-class rank \(\varrho=\varrho_p\). Suppose \(f=q\) is a regular \(p\)-admissible prime conductor for \(K\). Then the heterogeneous multiplet \(\mathbf{M}(K_{p,f})\) associated with the \(p\)-ring class field \(K_{p,f}\) mod \(f\) of \(K\) consists of two homogeneous multiplets with multiplicities \(m_p(K,1)\) and \(m_p(K,q)\). In this order, and in dependence on the \(p\)-ring space \(V_p(q)\), these two multiplicities are given by

• 1. \((1+p+\ldots+p^{\varrho-1},\ p^\varrho)\), \quad if \(V_p(q)=V\) (free situation),
• 2. \((1+p+\ldots+p^{\varrho-1},\ 0)\), \quad if \(V_p(q)< V\) (restrictive situation).

Proof. See [25, Theorem 3.2, p. 2215, and Theorem 3.3, p. 2217].

In the special case \(p=3\), there also exists the possibility of an irregular prime power conductor \(f=3^2\), provided the discriminant of the quadratic field satisfies the congruence \(d\equiv -3\,(\mathrm{mod}\,9)\).

Theorem 3. Assume that \(p=3\). Let \(K\) be a quadratic base field with \(3\)-class rank \(\varrho=\varrho_3\) and discriminant \(d\equiv -3\,(\mathrm{mod}\,9)\). Consider the irregular \(3\)-admissible prime power conductor \(f=3^2\) for \(K\). Then the heterogeneous multiplet \(\mathbf{M}(K_{p,f})\) associated with the \(3\)-ring class field \(K_{3,f}\) mod \(f\) of \(K\) consists of three homogeneous multiplets with multiplicities \(m_3(K,1)\), \(m_3(K,3)\) and \(m_3(K,9)\). In this order, and in dependence on the \(3\)-ring spaces \(V_3(3)\) and \(V_3(9)\), these three multiplicities are given by

• 1. \((1+3+\ldots+3^{\varrho-1},\ 3^\varrho,\ 3^{\varrho+1})\), if \(V_3(9)=V_3(3)=V\) (free situation),
• 2. \((1+3+\ldots+3^{\varrho-1},\ 3^\varrho,\ 0)\), if \(V_3(9)< V_3(3)=V\),
• 3. \((1+3+\ldots+3^{\varrho-1},\ 0,\ 3^\varrho)\), if \(V_3(9)=V_3(3)< V\),
• 4. \((1+3+\ldots+3^{\varrho-1},\ 0,\ 0)\), if \(V_3(9)< V_3(3)< V \) (maximal restriction).

Proof. See [25, Theorem 3.4, p. 2217].

5. Conductors with two prime divisors

For regular conductors \(f\) divisible by two primes, more distinct situations may arise.

Theorem 4. Let \(K\) be a quadratic base field with \(p\)-class rank \(\varrho=\varrho_p\). Suppose \(f=q_1\cdot q_2\) is a regular \(p\)-admissible conductor for \(K\) with two prime divisors \(q_1\) and \(q_2\). Then the heterogeneous multiplet \(\mathbf{M}(K_{p,f})\) associated with the \(p\)-ring class field \(K_{p,f}\) mod \(f\) of \(K\) consists of four homogeneous multiplets with multiplicities \(m_p(K,1)\), \(m_p(K,q_1)\), \(m_p(K,q_2)\) and \(m_p(K,f)\). In this order, and in dependence on the \(p\)-ring spaces \(V_p(q_1)\), \(V_p(q_2)\) and \(V_p(f)\), these four multiplicities are given by

• 1. \((1+p+\ldots+p^{\varrho-1},\ p^\varrho,\ p^\varrho,\ p^\varrho(p-1))\), if \(V_p(f)=V_p(q_1)=V_p(q_2)=V\) (free case),
• 2. \((1+p+\ldots+p^{\varrho-1},\ p^\varrho,\ 0,\ 0)\), if \(V_p(f)=V_p(q_2)< V_p(q_1)=V\),
• 3. \((1+p+\ldots+p^{\varrho-1},\ 0,\ p^\varrho,\ 0)\), if \(V_p(f)=V_p(q_1)< V_p(q_2)=V\),
• 4. \((1+p+\ldots+p^{\varrho-1},\ 0,\ 0,\ p^\varrho)\), if \(V_p(f)=V_p(q_1)=V_p(q_2)< V\),
• 5. \((1+p+\ldots+p^{\varrho-1},\ 0,\ 0,\ 0)\), if \(V_p(f)< V_p(q_1)\ne V_p(q_2)< V\) (maximal restriction).

Proof. We use the terminology and notation in [25]. Generally, the \(p\)-ring class rank is given by \(\varrho_{p,f}=\varrho+t+w-\delta_p(f)\). Here, we have either \(t=2\), \(w=0\) or \(t=1\), \(w=1\), and thus \(\varrho_{p,f}=\varrho+2-\delta_p(f)\). Also, we know that generally \(m_p(K,1)=\frac{p^\varrho-1}{p-1}\). Since \(f=q_1\cdot q_2\) is \(p\)-admissible, \(q_1\) and \(q_2\) must also be \(p\)-admissible, both.

• 1. In the free case with defect \(\delta_p(f)=0\), we have \(V_p(f)=V_p(q_1)=V_p(q_2)=V\) and \[ \frac{p^{\varrho+2}-1}{p-1}-\frac{p^\varrho-1}{p-1} =\frac{p^\varrho(p^2-1)}{p-1} =p^\varrho(p+1) =p^\varrho+p^\varrho+p^\varrho(p-1), \] which is exactly the desired partition \[ \frac{p^{\varrho_{p,f}}-1}{p-1}-m_p(K,1)=m_p(K,q_1)+m_p(K,q_2)+m_p(K,f). \]
• 2. If \(q_1\) is free and \(q_2\), \(f\) are restrictive, then \(V_p(f)=V_p(q_2)< V_p(q_1)=V\) and the relation \[ \frac{p^{\varrho+1}-1}{p-1}-\frac{p^\varrho-1}{p-1} =\frac{p^\varrho(p-1)}{p-1} =p^\varrho, \] must be interpreted as \(m_p(K,q_1)=p^\varrho\) and \(m_p(K,q_2)=m_p(K,f)=0\).
• 3. This case arises by interchanging the roles of \(q_1\) and \(q_2\) in the previous case.
• 4. Additionally to (2) and (3), there is another case of defect \(\delta_p(f)=1\) where neither \(q_1\) nor \(q_2\) is free but their \(p\)-ring spaces coincide \(V_p(f)=V_p(q_1)=V_p(q_2)< V\). Then the formula in (2) has to be interpreted as \(m_p(K,q_1)=m_p(K,q_2)=0\) and \(m_p(K,f)=p^\varrho\).
• 5. Finally, in the case of maximal restriction with defect \(\delta_p(f)=2\), which occurs for distinct \(p\)-ring spaces \(V_p(f)< V_p(q_1)\ne V_p(q_2)< V\), there is no rank increment from \(\varrho\) to \(\varrho_{p,f}\), and thus \(m_p(K,q_1)=m_p(K,q_2)=m_p(K,f)=0\).

6. Construction of \(p\)-ring class fields

This section describes how the classification of non-trivial \(p\)-ring class fields is prepared by their construction and rigorous count. The intended class field theoretic illumination of the structure of heterogeneous multiplets \(\mathbf{M}(K_{p,f})=\lbrack(N_{c,1},\ldots,N_{c,m(c)})\rbrack_{c\mid f}\) associated with \(p\)-ring class fields \(K_{p,f}\) modulo \(p\)-admissible conductors \(f\) over quadratic fields \(K\) must pay primary attention to the \(p\)-class rank \(\varrho_p\) of the quadratic base fields \(K=\mathbb{Q}(\sqrt{d})\), since \(\varrho_p\) enters the formula for the multiplicities \(m(c)\). More precisely, since the existence of a torsion free fundamental unit \(\varepsilon>1\) in real quadratic fields \(K\) with \(d>0\), and the occurrence of the \(3\)-torsion unit \(\zeta_3\) in the particular imaginary quadratic field \(K\) with \(d=-3\) in the case \(p=3\), exerts a crucial impact on the codimension of \(p\)-ring spaces \(V_p(c)\), the invariant \(\varrho_p\) must rather be replaced by the \(p\)-Selmer rank \(\sigma_p\) of \(K\) which describes all \(p\)-virtual units of \(K\), those which arise from non-trivial \(p\)-classes and the units in the usual sense: The secondary attention is devoted to various \(p\)-admissible conductors \(f=q_1\cdots q_t\) with an increasing number \(t\ge 0\) of prime divisors, starting with unramified extensions having \(t=0\), \(f=1\), and continuing with ramified extensions, beginning with prime or prime power conductors having \(t=1\), \(f=q_1\) with a prime \(q_1\in\mathbb{P}\) or the critical prime power \(q_1=p^2\).

7. Multiplets over imaginary quadratic fields for \(p=3\)

The focus of this section and most of the further sections is on \(p=3\), where the components \(N_{c,i}\) of multiplets are cyclic cubic extensions of quadratic base fields \(K\). Here, we begin with imaginary base fields \(K\) having the smallest possible \(3\)-Selmer rank \(\sigma_3=\varrho_3\). The behavior of the particular imaginary quadratic field \(K\) with \(d=-3\) where the extensions \(N_{c,i}/K\) contain pure cubic fields is rather similar to real quadratic base fields \(K\) with \(\sigma_3=\varrho_3+1\), and thus the case \(d=-3\) will be treated separately.

Theorem 5. Let \(K\) be an imaginary quadratic field with fundamental discriminant \(d< -3\) and trivial \(3\)-class rank \(\varrho_3=0\). Assume that \(f=q_1\cdots q_{\tau}\) is a \(3\)-admissible conductor with \(\tau\ge 1\) regular prime or prime power divisors \(q_i\) (that is, either \(q_i\equiv\pm 1\,(\mathrm{mod}\,3)\) or \(q_{\tau}=3\), \(d\equiv\pm 3\,(\mathrm{mod}\,9)\) or \(q_{\tau}=9\), \(d\equiv\pm 1\,(\mathrm{mod}\,3)\) but not \(q_{\tau}=9\), \(d\equiv -3\,(\mathrm{mod}\,9)\)). Then the \(3\)-ring class field \(K_{3,f}\) modulo \(f\) of \(K\) contains a homogeneous multiplet \(\mathbf{M}(K_{3,f})=(N_{f,1},\ldots,N_{f,m})\) of dihedral fields with conductor \(f\) and multiplicity \(m=2^{\tau-1}\) (singlet, doublet, quartet, octet, hexadecuplet, etc.).

Proof. All \(3\)-ring spaces \(V_3(q_i)\) coincide with \(3\)-Selmer space \(V=V_3\) [25, Theorem 3.2, p. 2215].

7.1. Classification of Angell's \(3169\) simply real cubic fields

In order to demonstrate the powerful performance of our innovative techniques, we construct all \(3\)-ring class fields \(K_{3,f}\) which contain the normal closures \(N\) of the simply real cubic fields \(L\) in Angell's table [11,12] as abelian extensions of the associated imaginary quadratic base fields \(K< N\).

There arise four values of the multiplicity \(m=1,2,3,4\), and accordingly simply real cubic fields are collected in singlets, doublets, triplets and quartets. Nilets with \(m=0\) complete the view.

The classification of the pure cubic fields, respectively non-pure simply real cubic fields, into differential principal factorization types was established in [28], respectively [9].

Although the types \(\alpha\) and \(\beta\) of pure cubic fields are similar to the types \(\alpha_2\) and \(\beta\) of non-pure simply real cubic fields, we do not mix the classifications, since firstly the existence of radicals among the principal factors distinguishes pure cubic fields from non-pure simply real cubic fields, and secondly, type \(\gamma\) can only occur for the former, whereas type \(\alpha_1\) is only possible for the latter.

Results

According to Table 1, the number of all non-pure simply real cubic fields \(L\) having discriminants \(-2\cdot 10^4< d_L< 0\) is given by \(\mathbf{3134}\). Together with \(35\) pure cubic fields in Table 2, the total number is \(\mathbf{3169}\), as announced correctly in [12].

Table 1. Cubic discriminants in the range \(-2\cdot 10^4< d_L=f^2\cdot d< 0\).

			Multiplicity					DPF
\(f\)	Condition	Total	\(0\)	\(1\)	\(2\)	\(3\)	\(4\)	\(\alpha_1\)	\(\alpha_2\)	\(\beta\)
\(q\)	\(\equiv -1\,(\mathrm{mod}\,3)\)	\(454\)	\(\)	\(454\)	\(\)	\(\)	\(\)	\(\)	\(\)	\(454\)
\(3\)	\(d\equiv +3\,(\mathrm{mod}\,9)\)	\(62\)	\(\)	\(62\)	\(\)	\(\)	\(\)	\(\)	\(\)	\(62\)
\(3\)	\(d\equiv -3\,(\mathrm{mod}\,9)\)	\(58\)	\(\)	\(58\)	\(\)	\(\)	\(\)	\(\)	\(\)	\(58\)
\(9\)	\(d\equiv -3\,(\mathrm{mod}\,9)\)	\(7\)	\(\)	\(\)	\(\)	\(7\)	\(\)	\(\)	\(\)	\(21\)
\(9\)	\(d\equiv -1\,(\mathrm{mod}\,3)\)	\(23\)	\(\)	\(23\)	\(\)	\(\)	\(\)	\(\)	\(\)	\(23\)
\(9\)	\(d\equiv +1\,(\mathrm{mod}\,3)\)	\(20\)	\(\)	\(20\)	\(\)	\(\)	\(\)	\(\)	\(16\)	\(4\)
\(\ell\)	\(\equiv +1\,(\mathrm{mod}\,3)\)	\(64\)	\(\)	\(64\)	\(\)	\(\)	\(\)	\(\)	\(49\)	\(15\)
\(q_1q_2\)	\(\equiv -1\,(\mathrm{mod}\,3)\)	\(6\)	\(\)	\(\)	\(6\)	\(\)	\(\)	\(\)	\(\)	\(12\)
\(3q\)	\(d\equiv +3\,(\mathrm{mod}\,9)\)	\(7\)	\(\)	\(\)	\(7\)	\(\)	\(\)	\(\)	\(\)	\(14\)
\(3q\)	\(d\equiv -3\,(\mathrm{mod}\,9)\)	\(3\)	\(\)	\(\)	\(3\)	\(\)	\(\)	\(\)	\(\)	\(6\)
\(9q\)	\(d\equiv -1\,(\mathrm{mod}\,3)\)	\(3\)	\(\)	\(\)	\(3\)	\(\)	\(\)	\(\)	\(\)	\(6\)
\(9q\)	\(d\equiv +1\,(\mathrm{mod}\,3)\)	\(3\)	\(\)	\(\)	\(3\)	\(\)	\(\)	\(\)	\(\)	\(6\)
\(3\ell\)	\(d\equiv +3\,(\mathrm{mod}\,9)\)	\(1\)	\(\)	\(\)	\(1\)	\(\)	\(\)	\(\)	\(\)	\(2\)
\(q\ell\)	\(\equiv\mp 1\,(\mathrm{mod}\,3)\)	\(1\)	\(\)	\(\)	\(1\)	\(\)	\(\)	\(\)	\(\)	\(2\)
\(1\)	\(\varrho_3=1\)	\(2143\)	\(\)	\(2143\)	\(\)	\(\)	\(\)	\(2143\)	\(\)	\(\)
\(q\)	\(\equiv -1\,(\mathrm{mod}\,3)\)	\(196\)	\(162\)	\(\)	\(\)	\(34\)	\(\)	\(87\)	\(\)	\(15\)
\(3\)	\(d\equiv +3\,(\mathrm{mod}\,9)\)	\(24\)	\(22\)	\(\)	\(\)	\(2\)	\(\)	\(4\)	\(\)	\(2\)
\(3\)	\(d\equiv -3\,(\mathrm{mod}\,9)\)	\(22\)	\(16\)	\(\)	\(\)	\(6\)	\(\)	\(13\)	\(\)	\(5\)
\(9\)	\(d\equiv -1\,(\mathrm{mod}\,3)\)	\(5\)	\(5\)	\(\)	\(\)	\(\)	\(\)	\(\)	\(\)	\(\)
\(9\)	\(d\equiv +1\,(\mathrm{mod}\,3)\)	\(9\)	\(8\)	\(\)	\(\)	\(1\)	\(\)	\(2\)	\(\)	\(1\)
\(\ell\)	\(\equiv +1\,(\mathrm{mod}\,3)\)	\(22\)	\(19\)	\(\)	\(\)	\(3\)	\(\)	\(7\)	\(\)	\(2\)
\(q_1q_2\)	\(\equiv -1\,(\mathrm{mod}\,3)\)	\(2\)	\(1\)	\(\)	\(\)	\(1\)	\(\)	\(\)	\(\)	\(3\)
\(3q\)	\(d\equiv +3\,(\mathrm{mod}\,9)\)	\(3\)	\(1\)	\(\)	\(\)	\(2\)	\(\)	\(\)	\(\)	\(6\)
\(9q\)	\(d\equiv +1\,(\mathrm{mod}\,3)\)	\(1\)	\(\)	\(\)	\(\)	\(1\)	\(\)	\(\)	\(\)	\(3\)
\(q\ell\)	\(\equiv\mp 1\,(\mathrm{mod}\,3)\)	\(2\)	\(1\)	\(\)	\(\)	\(1\)	\(\)	\(\)	\(\)	\(3\)
\(1\)	\(\varrho_3=2\)	\(22\)	\(\)	\(\)	\(\)	\(\)	\(22\)	\(88\)	\(\)	\(\)
	Summary	\(3163\)	\(235\)	\(2824\)	\(24\)	\(58\)	\(22\)	\(2344\)	\(65\)	\(725\)

We emphasize the difference between the number of discriminants (without multiplicities)

\[2824+24+58+22=2928,\] and the number of fields (including multiplicities in a weighted sum) \[1\cdot 2824+2\cdot 24+3\cdot 58+4\cdot 22=2824+48+174+88=3134,\] which can be confirmed by adding the contributions to the \(3\) DPF types \(\alpha_1\), \(\alpha_2\), \(\beta\) \[2344+65+725=3134.\] In contrast, \(235\) is the number of formal cubic discriminants \(d_L=f^2\cdot d_K\) with fundamental discriminants \(d_K\) of imaginary quadratic fields and \(3\)-admissible conductors \(f\) for each \(K\), where the relevant multiplicity formula [25] yields the value zero. So the formal cubic discriminants belong to nilets, i.e., multiplets with multiplicity \(m_3(K,f)=0\). The total number of all (actual) cubic discriminants and formal cubic discriminants is the number of admissible cubic discriminants \[2928+235=3163.\] According to Theorem 5, Nilets can only arise for \(\varrho_3\ge 1\), but not for \(\varrho_3=0\).

Table 2. Pure cubic discriminants in the range \(-2\cdot 10^4< d_L=-3\cdot f^2< 0\).

			Multiplicity					DPF
\(f\)	Condition	Total	\(0\)	\(1\)	\(2\)	\(3\)	\(4\)	\(\alpha\)	\(\beta\)	\(\gamma\)
\(q\)	\(\equiv -1\,(\mathrm{mod}\,3)\)	\(11\)	\(8\)	\(3\)	\(\)	\(\)	\(\)	\(\)	\(\)	\(3\)
\(9\)	\(d=-3\)	\(1\)	\(\)	\(1\)	\(\)	\(\)	\(\)	\(\)	\(\)	\(1\)
\(\ell\)	\(\equiv +1\,(\mathrm{mod}\,3)\)	\(10\)	\(7\)	\(3\)	\(\)	\(\)	\(\)	\(3\)	\(\)	\(\)
\(q_1q_2\)	\(\equiv -1\,(\mathrm{mod}\,3)\)	\(6\)	\(1\)	\(5\)	\(\)	\(\)	\(\)	\(\)	\(5\)	\(\)
\(3q\)	\(d=-3\)	\(5\)	\(1\)	\(4\)	\(\)	\(\)	\(\)	\(\)	\(4\)	\(\)
\(9q\)	\(d=-3\)	\(2\)	\(\)	\(\)	\(2\)	\(\)	\(\)	\(\)	\(4\)	\(\)
\(3\ell\)	\(d=-3\)	\(3\)	\(1\)	\(2\)	\(\)	\(\)	\(\)	\(2\)	\(\)	\(\)
\(9\ell\)	\(d=-3\)	\(1\)	\(\)	\(\)	\(1\)	\(\)	\(\)	\(2\)	\(\)	\(\)
\(q\ell\)	\(\equiv\mp 1\,(\mathrm{mod}\,3)\)	\(8\)	\(2\)	\(6\)	\(\)	\(\)	\(\)	\(4\)	\(2\)	\(\)
\(q_1q_2\ell\)	\(\equiv\mp 1\,(\mathrm{mod}\,3)\)	\(1\)	\(\)	\(1\)	\(\)	\(\)	\(\)	\(\)	\(1\)	\(\)
\(3q_1q_2\)	\(d=-3\)	\(2\)	\(\)	\(2\)	\(\)	\(\)	\(\)	\(\)	\(2\)	\(\)
\(3q\ell\)	\(d=-3\)	\(2\)	\(\)	\(2\)	\(\)	\(\)	\(\)	\(\)	\(2\)	\(\)
	Summary	\(52\)	\(20\)	\(29\)	\(3\)	\(\)	\(\)	\(11\)	\(20\)	\(4\)

According to Table 2, the number of pure cubic fields \(L\) with discriminant \(-2\cdot 10^4< d_L< 0\) is \(35\). Actually, triplets and quartets of pure cubic fields do not occur in this range.

There is a difference between the number of discriminants (without multiplicities) \[29+3=32,\] and the number of fields (including multiplicities in a weighted sum) \[1\cdot 29+2\cdot 3=29+6=35,\] which can be confirmed by adding the contributions to the \(3\) DPF types \[11+20+4=35.\] The total number of all (actual) cubic discriminants and formal cubic discriminants (of the \(20\) nilets) is the number of admissible pure cubic discriminants \(d_L=-3\cdot f^2\), \[32+20=52.\]

8. Multiplets over real quadratic fields for \(p=3\)

We continue with real quadratic base fields \(K\) having elevated \(3\)-Selmer rank \(\sigma_3=\varrho_3+1\), due to the existence of a torsion free fundamental unit \(\varepsilon>1\).

8.1. Classification of Angell's \(4804\) totally real cubic fields

In order to demonstrate our progressive perspective of classification of heterogeneous multiplets \(\mathbf{M}(K_{3,f})\) into an enigmatic variety of differential principal factorization types, we construct all \(3\)-ring class fields \(K_{3,f}\) which contain the normal closures \(N\) of the totally real cubic fields \(L\) in Angell's table [13,14] as abelian extensions of the associated real quadratic base fields \(K< N\).

Again there arise four values of the multiplicity \(m=1,2,3,4\), and accordingly totally real cubic fields are collected in singlets, doublets, triplets and quartets. Formal nilets complete the view.

The classification into differential principal factorization types for non-cyclic totally real cubic fields was developed in [9,26,27].

Results

According to Table 3, the number of non-cyclic totally real cubic fields \(L\) with discriminant \(0< d_L< 10^5\) is \(\mathbf{4753}\), in perfect accordance with the results by Llorente and Oneto [32,33], who discovered the ommission of ten fields in the table by Angell [13,14]. Together with \(51\) cyclic cubic fields in Table 4, the total number is \(\mathbf{4804}\) (not \(4794\), as announced erroneously in [14]).

Again we emphasize the difference between the number of discriminants (without multiplicities)

\[4652+9+21+5=4687,\] and the number of fields (including multiplicities in a weighted sum) \[1\cdot 4652+2\cdot 9+3\cdot 21+4\cdot 5=4652+18+63+20=4753,\] which can be confirmed by adding the contributions to the \(7\) DPF types (\(\alpha_2\), \(\alpha_3\) do not occur) \[16+10+76+106+3349+79+1117=4753.\]

Table 3. Cubic discriminants in the range \(0< d_L=f^2\cdot d< 10^5\).

			Multiplicity					Differential Principal Factorization
\(f\)	Condition	Total	\(0\)	\(1\)	\(2\)	\(3\)	\(4\)	\(\alpha_1\)	\(\beta_1\)	\(\beta_2\)	\(\gamma\)	\(\delta_1\)	\(\delta_2\)	\(\varepsilon\)
\(q\)	\(\equiv -1\,(\mathrm{mod}\,3)\)	\(3025\)	\(2219\)	\(806\)	\(\)	\(\)	\(\)	\(\)	\(\)	\(\)	\(\)	\(\)	\(\)	\(806\)
\(3\)	\(d\equiv +3\,(\mathrm{mod}\,9)\)	\(396\)	\(287\)	\(109\)	\(\)	\(\)	\(\)	\(\)	\(\)	\(\)	\(\)	\(\)	\(\)	\(109\)
\(3\)	\(d\equiv -3\,(\mathrm{mod}\,9)\)	\(389\)	\(284\)	\(105\)	\(\)	\(\)	\(\)	\(\)	\(\)	\(\)	\(\)	\(\)	\(\)	\(105\)
\(9\)	\(d\equiv -3\,(\mathrm{mod}\,9)\)	\(48\)	\(9\)	\(38\)	\(\)	\(1\)	\(\)	\(\)	\(\)	\(\)	\(\)	\(\)	\(\)	\(41\)
\(9\)	\(d\equiv -1\,(\mathrm{mod}\,3)\)	\(136\)	\(102\)	\(34\)	\(\)	\(\)	\(\)	\(\)	\(\)	\(\)	\(\)	\(\)	\(\)	\(34\)
\(9\)	\(d\equiv +1\,(\mathrm{mod}\,3)\)	\(127\)	\(96\)	\(31\)	\(\)	\(\)	\(\)	\(\)	\(\)	\(8\)	\(\)	\(\)	\(20\)	\(3\)
\(\ell\)	\(\equiv +1\,(\mathrm{mod}\,3)\)	\(402\)	\(316\)	\(86\)	\(\)	\(\)	\(\)	\(\)	\(\)	\(20\)	\(\)	\(\)	\(59\)	\(7\)
\(q_1q_2\)	\(\equiv -1\,(\mathrm{mod}\,3)\)	\(70\)	\(30\)	\(38\)	\(2\)	\(\)	\(\)	\(\)	\(\)	\(\)	\(38\)	\(\)	\(\)	\(4\)
\(3q\)	\(d\equiv +3\,(\mathrm{mod}\,9)\)	\(46\)	\(23\)	\(23\)	\(\)	\(\)	\(\)	\(\)	\(\)	\(\)	\(23\)	\(\)	\(\)	\(\)
\(3q\)	\(d\equiv -3\,(\mathrm{mod}\,9)\)	\(45\)	\(19\)	\(25\)	\(1\)	\(\)	\(\)	\(\)	\(\)	\(\)	\(25\)	\(\)	\(\)	\(2\)
\(9q\)	\(d\equiv -3\,(\mathrm{mod}\,9)\)	\(5\)	\(\)	\(\)	\(4\)	\(1\)	\(\)	\(\)	\(\)	\(\)	\(9\)	\(\)	\(\)	\(2\)
\(9q\)	\(d\equiv -1\,(\mathrm{mod}\,3)\)	\(14\)	\(6\)	\(8\)	\(\)	\(\)	\(\)	\(\)	\(\)	\(\)	\(8\)	\(\)	\(\)	\(\)
\(9q\)	\(d\equiv +1\,(\mathrm{mod}\,3)\)	\(15\)	\(5\)	\(10\)	\(\)	\(\)	\(\)	\(\)	\(\)	\(10\)	\(\)	\(\)	\(\)	\(\)
\(9\ell\)	\(d\equiv -1\,(\mathrm{mod}\,3)\)	\(1\)	\(\)	\(1\)	\(\)	\(\)	\(\)	\(\)	\(\)	\(1\)	\(\)	\(\)	\(\)	\(\)
\(3\ell\)	\(d\equiv +3\,(\mathrm{mod}\,9)\)	\(6\)	\(1\)	\(5\)	\(\)	\(\)	\(\)	\(\)	\(\)	\(5\)	\(\)	\(\)	\(\)	\(\)
\(3\ell\)	\(d\equiv -3\,(\mathrm{mod}\,9)\)	\(5\)	\(2\)	\(3\)	\(\)	\(\)	\(\)	\(\)	\(\)	\(3\)	\(\)	\(\)	\(\)	\(\)
\(q\ell\)	\(\equiv\mp 1\,(\mathrm{mod}\,3)\)	\(43\)	\(13\)	\(29\)	\(1\)	\(\)	\(\)	\(\)	\(\)	\(29\)	\(\)	\(\)	\(\)	\(2\)
\(3q_1q_2\)	\(d\equiv +3\,(\mathrm{mod}\,9)\)	\(2\)	\(\)	\(1\)	\(1\)	\(\)	\(\)	\(\)	\(\)	\(\)	\(3\)	\(\)	\(\)	\(\)
\(1\)	\(\varrho_3=1\)	\(3300\)	\(\)	\(3300\)	\(\)	\(\)	\(\)	\(\)	\(\)	\(\)	\(\)	\(3300\)	\(\)	\(\)
\(q\)	\(\equiv -1\,(\mathrm{mod}\,3)\)	\(275\)	\(261\)	\(\)	\(\)	\(14\)	\(\)	\(\)	\(4\)	\(\)	\(\)	\(36\)	\(\)	\(2\)
\(3\)	\(d\equiv -3\,(\mathrm{mod}\,9)\)	\(35\)	\(34\)	\(\)	\(\)	\(1\)	\(\)	\(\)	\(\)	\(\)	\(\)	\(3\)	\(\)	\(\)
\(\ell\)	\(\equiv +1\,(\mathrm{mod}\,3)\)	\(28\)	\(25\)	\(\)	\(\)	\(3\)	\(\)	\(\)	\(3\)	\(\)	\(\)	\(6\)	\(\)	\(\)
\(3q\)	\(d\equiv -3\,(\mathrm{mod}\,9)\)	\(2\)	\(1\)	\(\)	\(\)	\(1\)	\(\)	\(\)	\(3\)	\(\)	\(\)	\(\)	\(\)	\(\)
\(1\)	\(\varrho_3=2\)	\(5\)	\(\)	\(\)	\(\)	\(\)	\(5\)	\(16\)	\(\)	\(\)	\(\)	\(4\)	\(\)	\(\)
	Summary	\(8420\)	\(3733\)	\(4652\)	\(9\)	\(21\)	\(5\)	\(16\)	\(10\)	\(76\)	\(106\)	\(3349\)	\(79\)	\(1117\)

In contrast, \(3733\) is the number of formal cubic discriminants \(d_L=f^2\cdot d_K\) with fundamental discriminants \(d_K\) of real quadratic fields and \(3\)-admissible conductors \(f\) for each \(K\), where the relevant multiplicity formula [25] yields the value zero. So the formal cubic discriminants belong to nilets, i.e., multiplets with multiplicity \(m_3(K,f)=0\). The total number of all (actual) cubic discriminants and formal cubic discriminants is the number of admissible cubic discriminants

\[4687+3733=8420.\]

Table 4. Cyclic cubic discriminants in the range \(0< d_L=f^2< 10^5\).

		M		DPF
\(f\)	Condition	\(1\)	\(2\)	\(\zeta\)
\(9\)	\(d=1\)	\(1\)	\(\)	\(1\)
\(\ell\)	\(\equiv +1\,(\mathrm{mod}\,3)\)	\(30\)	\(\)	\(30\)
\(9\ell\)	\(d=1\)	\(\)	\(4\)	\(8\)
\(\ell_1\ell_2\)	\(\equiv +1\,(\mathrm{mod}\,3)\)	\(\)	\(6\)	\(12\)
	Summary	\(31\)	\(10\)	\(51\)

According to Table 4, the number of cyclic cubic fields \(L\) with discriminant \(0< d_L< 10^5\) is \(51\), with \(31\) arising from singlets having conductors \(f\) with a single prime divisor, and \(20\) from doublets having two prime divisors of the conductor \(f\). (M denotes the multiplicity.)

We point out that cyclic cubic fields are rather contained in ray class fields over \(\mathbb{Q}\) than in ring class fields over real quadratic base fields. The single possible DPF type \(\zeta\) has nothing to do with the \(9\) DPF types \(\alpha_1,\alpha_2,\alpha_3,\beta_1,\beta_2,\gamma,\delta_1,\delta_2,\varepsilon\) of non-abelian totally real cubic fields in [9].

9. Conclusion and outlook

In this paper, we have classified all multiplets \(\mathrm{Obj}(K_{3,f})\) of non-pure simply real cubic fields \(L\) (more precisely of their normal closures \(N\)) according to the associated multiplets of invariants, namely the differential principal factorization types, \(\mathrm{Inv}(K_{3,f})\), where \(K_{3,f}\) denotes the \(3\)-ring class field modulo a \(3\)-admissible conductor \(f\) of the imaginary quadratic subfield \(K< N\): (Recall that \(\mathrm{Obj}(K_{3,f})=(N_{f,i})_{1\le i\le m}\) and \(\mathrm{Inv}(K_{3,f})=(\tau(N_{f,i}))_{1\le i\le m}\), here homogeneously.)

• \(2824\) singlets of type either \((\alpha_1)\) or \((\alpha_2)\) or \((\beta)\), according to Table 1;
• \(24\) doublets of exclusive type \((\beta,\beta)\) (without \(3\) pure cubic doublets);
• \(58\) triplets with the following distribution of types:
• -   \(7\) triplets of type \((\beta,\beta,\beta)\) for \(f=9\) singular, \(\varrho_3=0\),
• -   \(51\) triplets sharing common \(3\)-class rank \(\varrho_3=1\) of \(K\) [34, Table 1, pp. 118-121], namely
• *   \(34\) triplets of type \((\alpha_1,\alpha_1,\alpha_1)\) for \(f=q,\ell,3\),
• *   \(3\) triplets of type \((\alpha_1,\alpha_1,\beta)\) for \(f=\ell,9\) split,
• *   \(5\) triplets of type \((\alpha_1,\beta,\beta)\) for \(f=q,3\), and
• *   \(9\) triplets of type \((\beta,\beta,\beta)\) for \(f=q,3,3q,q\ell,9q\), finer than [34] since \(\alpha_2\) does not occur;
• \(22\) quartets of exclusive type \((\alpha_1,\alpha_1,\alpha_1,\alpha_1)\) (see [35] for details concerning the capitulation).

Similarly, we have classified all multiplets \(\mathrm{Obj}(K_{3,f})\) of non-cyclic totally real cubic fields \(L\) (more precisely of their normal closures \(N\)) according to the associated multiplets of invariants, namely the differential principal factorization types, \(\mathrm{Inv}(K_{3,f})\), where \(K_{3,f}\) denotes the \(3\)-ring class field modulo a \(3\)-admissible conductor \(f\) of the real quadratic subfield \(K< N\):

• \(4652\) singlets of type either \((\beta_2)\) or \((\gamma)\) or \((\delta_1)\) or \((\delta_2)\) or \((\varepsilon)\), according to Table 3;
• \(9\) doublets, \(4\) of type \((\gamma,\gamma)\) for \(f=9q,3q_1q_2\) and \(5\) of type \((\varepsilon,\varepsilon)\) for \(f=3q,9q,q_1q_2,q\ell\);
• \(21\) triplets with the following distribution of types:
• -   \(1\) triplet of type \((\varepsilon,\varepsilon,\varepsilon)\) for \(f=9\) singular, \(\varrho_3=0\),
• -   \(1\) triplet of type \((\gamma,\gamma,\gamma)\) for \(f=9q\) singular, \(\varrho_3=0\), and
• -   \(19\) triplets sharing common \(3\)-class rank \(\varrho_3=1\) of \(K\) (with considerable refinement of Schmithals' coarse distinction of only two alternatives [34, Table 2, pp. 122-123]), namely
• *   \(13\) triplets of type \((\delta_1,\delta_1,\delta_1)\) for \(f=3,q\),
• *   \(1\) triplet of type \((\beta_1,\beta_1,\beta_1)\) for \(f=3q\),
• *   \(2\) triplets of type \((\beta_1,\beta_1,\varepsilon)\) for \(f=q\) (conspicuously with symbol ''-'' in [34]), and
• *   \(3\) triplets of type \((\beta_1,\delta_1,\delta_1)\) for \(f=\ell\),
• \(5\) quartets, \(1\) of type \((\alpha_1,\alpha_1,\alpha_1,\alpha_1)\) and \(4\) of type \((\alpha_1,\alpha_1,\alpha_1,\delta_1)\) (more details in [36,37]).

In the same manner, we shall refine more extensive tables by Fung and Williams [18], Ennola and Turunen [15,16], Llorente and Quer [17] in the new year \(2021\).

Moreover, we shall provide extensive evidence of the truth of Scholz' conjecture, which we have proved for \(p=3\) in [9], also for \(p=5\) and \(p=7\), and probably for any odd prime \(p\).

Data Availability

Implementations of our innovative algorithms in Magma [5,6,7] may be requested via email.

Acknowledgments

This work has been completed on Tuesday, 29 December 2020. In order to disprove any claims of priority concerning the innovative perspective of classifying multiplets of dihedral fields, contained in \(p\)-ring class fields, into differential principal factorization types, the article has immediately been disseminated on various scientific open access platforms.

Funding Information

The author gratefully acknowledges that his research was supported by the Austrian Science Fund (FWF): projects J0497-PHY and P26008-N25.

Conflicts of Interests

The author declares no conflict of interest.