1 Introduction and Main Theorem↩︎

Let \(x^n-a\) be an irreducible polynomial in \(\mathbb{Z}[x]\). The primary objective of this paper is to establish Theorem 1, an explicit description of the splitting of the integral prime 2 in \({\mathcal{O}}_{\mathbb{Q}(\sqrt[n]{a})}\) the ring of integers of \(\mathbb{Q}\big(\sqrt[n]{a}\big)\). Combining this with Theorem 2, the result for odd primes proved in [1], we achieve a complete, explicit description of prime splitting in radical¹ extensions of \(\mathbb{Q}\). Numerous examples of these theorems are computed in Section 5, and a SageMath implementation can be found here: https://doi.org/10.5281/zenodo.20208664.

In Corollaries 1 and 2, this description is employed to classify the common \(N\)-index divisors (primes dividing the index of any order \(\mathbb{Z}\big[\theta_1,\dots, \theta_N\big]\) in the maximal order \({\mathcal{O}}_K\)) of a radical extension. Using this we can construct radical extensions with common \(N\)-index divisors for arbitrary \(N\); see Example 1. In Proposition 8, we construct non-radical extensions of minimal degree having a common \(N\)-index divisor. In the other direction, in Section 6 we use the index form to construct extensions that are not monogenic yet have no common index divisors. Before we state our theorems, we make the following definition. Let \(L/K\) be an extension of number fields with respective rings of integers \({\mathcal{O}}_L\) and \({\mathcal{O}}_K\). Let \({\mathfrak{p}}\subset {\mathcal{O}}_K\) be a prime ideal with residue field \(k_{\mathfrak{p}}\mathrel{\vcenter{:}}= {\mathcal{O}}_K/{\mathfrak{p}}\). An ideal \(\mathfrak{I}\subset {\mathcal{O}}_L\) mirrors the factorization of a polynomial \(f(x)\in k_{\mathfrak{p}}[x]\) if, given a factorization into irreducibles \(f(x)=\prod_{i=1}^r \phi_i(x)^{e_i}\) in \(k_{\mathfrak{p}}[x]\), there is a prime ideal factorization \(\mathfrak{I}=\prod_{i=1}^r{\mathfrak{P}}_i^{e_i}\) in \({\mathcal{O}}_L\) where the degree of \({\mathfrak{P}}_i\) over \({\mathfrak{P}}_i\cap {\mathcal{O}}_K\), i.e. \([{\mathcal{O}}_L/{\mathfrak{P}}_i:k_{\mathfrak{p}}]\), is equal to the degree of \(\phi_i(x)\).

We can now state our main theorem.

The complement to Theorem 1 for odd \(p\) is [1]. We restate it here.

As an application, we classify the common \(N\)-index divisors of \(\mathbb{Q}\big(\sqrt[n]{a}\big)\). Fix \(N>0\) and let \(K\) be a number field. We say a prime \(p\) is a common \(N\)-index divisor, or C\(N\)-ID for brevity, if for any \(\big(\theta_1, \dots, \theta_N\big)\in {\mathcal{O}}_K^N\) the index of \(\mathbb{Z}\big[\theta_1,\dots, \theta_N\big]\) in \({\mathcal{O}}_K\) is divisible by \(p\) (or infinite in the case where \(\mathbb{Q}\big(\theta_1,\dots, \theta_N\big)\neq K\)). In particular, if \(p\) is a common \(N\)-index divisor, then at least \(N+1\) ring generators are required to generate \({\mathcal{O}}_K\) over \(\mathbb{Z}\).

For \(N=1\), these are the primes whose splitting prevents monogenicity. Indeed the first non-monogenic field was constructed by Dedekind [2]. He showed that in the cubic field given by \(x^3-x^2-2x-8\), the prime 2 is a common index² divisor.

The work of Pleasants [5] building on the work of Hensel [6] and Dedekind [2] for \(N=1\) shows that common \(N\)-index divisors can be classified from splitting information. Indeed, Pleasants’ main theorem (Theorem 7) allows us to use Theorems 1 and 2 to fully understand common \(N\)-index divisors in radical extensions.

First, we let \(q=p^N\) and recall that Gauss’s formula for the number of monic irreducible polynomials of degree \(f\) over \(\mathbb{F}_{q}\) is \[\operatorname{Irred}(f,q)\mathrel{\vcenter{:}}= \frac{1}{f}\sum_{d\mid f}\mu\left(\frac{f}{d}\right)q^d, \text{ where } \mu \text{ is the M\"obius function.}\]

For odd primes \(p\), the following result generalizes the classification of common index divisors in [1]:

1.1 Organization↩︎

Above we have stated the main theorem (Theorem 1), its complement for odd \(p\) (Theorem 2 proved in [1]), and the respective corollaries describing common \(N\)-index divisors (Corollaries 1 and 2). Despite almost verbatim overlap with [1], for the convenience of the reader, we undertake a brief summary of previous work (Subsection 1.2) and a description of the initial dissections of the Montes algorithm/Ore’s factorization theorem (Section 2). Section 3 is devoted to the proof of Theorem 1. The foundation of this proof is [7], Vélez’s classification of the splitting of 2 in \(\mathbb{Q}\big(\sqrt[2^m]{a}\big)\). Section 4 undertakes the proof of Corollaries 1 and 2. Pleasants’s result [5] on the connection between prime splitting and common \(N\)-index divisors is key here. Novel examples of number rings with common \(N\)-index divisors are included as well; see Example 1 and Proposition 8. Section 5 is devoted to a number of examples of Theorem 1. Section 6 gives a new construction of non-monogenic fields without common index divisors, exceptional number fields in Pleasants’s terminology. The construction is distinctive in that it uses prescribed vanishing of terms in the index form to create exceptional radical extensions of arbitrary degree \(n>2\).

1.2 Previous work↩︎

There has been a good deal of recent work on common 1-index divisors. See [8], [9], [10], [11], [12], [13], [14], as well as other works from these authors for various families of polynomials/fields of a fixed degree. When consulting the literature, it bears noting that the question of whether or not a given a binomial (radical) polynomial or a trinomial polynomial is monogenic has been fully answered. See [15] and [16], respectively. For families of varying degree, [17] study multiquadratic fields, while [18] study fields generated by polynomials of prime degree with dihedral Galois groups.

[4], the recent book by Gouvêa and Webster, provides a vibrant historical account of the impact of common index divisors on the development of algebraic number theory. Further, the book includes new translations of two of the seminal papers already referenced: [2] and [6].

We note that [19] uses Newton polygon techniques to establish an integral basis for \(\mathbb{Q}( \sqrt[n]{a})\). In some cases, this work can describe the splitting of primes dividing \(na\), but it does not fully describe splitting. The general method is subsumed by earlier work of Ore [20]. In [21], the author computes bounds on the conductors of extensions obtained by a root of unity and a radical. In a particular case when \(p=2\), exact values of the conductor are computed. [22] constructs a uniformizer for \(\mathbb{Q}_p\big(\zeta_p, \sqrt[p^{i+1}]{a}\big)/\mathbb{Q}_p\big(\zeta_p, \sqrt[p^{i}]{a}\big)\). [23] describes a uniformizer for the extension \(\mathbb{Q}_p\big(\zeta_{p^2},\sqrt[p]{p}\big)/\mathbb{Q}_p\), while [24] continues down this avenue by constructing a uniformizer for \(\mathbb{Q}_p\big(\zeta_{p^2},\sqrt[p^m]{p}\big)/\mathbb{Q}_p\) with \(m\geq 1\).

In [25], Vélez describes the factorization of a prime above \(p\) in any extension of a number field obtained by adjoining a \(p^{\text{th}}\) power or a \(p^{\text{th}}\) root of unity. In the series of papers [26] and [27], prime ideal splitting in radical extensions is described in a variety of cases. In [7], Vélez completely describes the splitting of the prime \(p\) in a \(p\)-power radical extension of \(\mathbb{Q}\). This work will be our jumping off point (Theorem 5) in addressing the more difficult cases (Cases [MainIII] and [MainIV]) of Theorem 1. Vélez’s primary application was describing the genus field of \(\mathbb{Q}\big( \sqrt[n]{a}\big)\), while our application is the classification of common \(N\)-index divisors of \(\mathbb{Q}\big( \sqrt[n]{a}\big)\). Thus, we have rephrased and slightly reinterpreted Vélez’s results in order to suit our goals.

2 The Montes algorithm and a theorem of Ore↩︎

The Montes algorithm is a \(p\)-adic factorization algorithm that is based on and extends the pioneering work of Øystein Ore [28]. We will essentially only employ the aspects developed by Ore here, but we will use the notation and setup of the general implementation. For the complete development of the Montes algorithm, see [29]. Our notation will roughly follow [30], which gives a more extensive summary than we undertake here. One can also consult [31].

Let \(p\) be an integral prime, \(K\) a number field with ring of integers \({\mathcal{O}}_K\), and \({\mathfrak{p}}\) a prime of \(K\) above \(p\). Write \(K_{\mathfrak{p}}\) to denote the completion of \(K\) at \({\mathfrak{p}}\). By a uniformizer at \({\mathfrak{p}}\) or a uniformizer of \(K_{\mathfrak{p}}\), we mean an element \(\pi_{\mathfrak{p}}\in{\mathcal{O}}_K\) such that \(v_{\mathfrak{p}}\big(\pi_{\mathfrak{p}}\big)=1\). Suppose we have a monic, irreducible polynomial \(f(x)\in {\mathcal{O}}_K[x]\). We extend the standard \({\mathfrak{p}}\)-adic valuation to \({\mathcal{O}}_K[x]\) by defining the \({\mathfrak{p}}\)-adic valuation of \(f(x) = a_n x^n + \cdots + a_1 x + a_0 \in {\mathcal{O}}_K[x]\) to be \[v_{\mathfrak{p}}\big(f(x)\big) = \min_{0 \leq i \leq n} \big( v_{\mathfrak{p}}(a_i) \big).\] This is sometimes called the Gauss valuation. If \(\phi(x), f(x) \in {\mathcal{O}}_K[x]\) are monic and such that \(\deg \phi \leq \deg f\), then we can write \[f(x)=\sum_{i=0}^k a_i(x)\phi(x)^i,\] for some \(k\), where each \(a_i(x) \in {\mathcal{O}}_K[x]\) has degree less than \(\deg \phi\). We call the above expression the \(\phi\)-adic development of \(f(x)\). We associate to the \(\phi\)-adic development of \(f(x)\) an open Newton polygon by taking the lower convex hull of the integer lattice points \(\big(i,v_p(a_i(x))\big)\). The sides of the Newton polygon with negative slope are the principal \(\phi\)-polygon.

Write \(k_{\mathfrak{p}}\) for the residue field \({\mathcal{O}}_K/{\mathfrak{p}}\), and let \(\overline{f(x)}\) be the image of \(f(x)\) in \(k_{\mathfrak{p}}[x]\). It will often be the case that we develop \(f(x)\) with respect to an irreducible factor \(\phi(x)\) of \(\overline{f(x)}\). In this situation, we will want to consider the extension of \(k_{\mathfrak{p}}\) obtained by adjoining a root of \(\phi(x)\). We denote this finite field by \(k_{{\mathfrak{p}},\phi}\). We associate to each side of the principal \(\phi\)-polygon a polynomial in \(k_{{\mathfrak{p}},\phi}[y]\). Suppose \(S\) is a side of the principal \(\phi\)-polygon with initial vertex \(\big(s,v_{\mathfrak{p}}(a_s(x))\big)\), terminal vertex \(\big(k,v_{\mathfrak{p}}(a_k(x))\big)\), and slope \(-\frac{h}{e}\) written in lowest terms. Define the length of the side to be \(l(S)=k-s\) and the degree to be \(d\mathrel{\vcenter{:}}=\frac{l(S)}{e}\). Let \(\operatorname{red}:{\mathcal{O}}_K[x]\to k_{{\mathfrak{p}},\phi}\) denote the homomorphism obtained by quotienting by the ideal \(\big({\mathfrak{p}},\phi(x)\big)\). For each \(i\) in the range \(b\leq i\leq k\), we define the residual coefficient to be \[c_i=\left\{ \begin{array}{ll} 0 \text{ if } \big(i,v_{\mathfrak{p}}(a_i(x))\big) \text{ lies strictly above } S \text{ or } v_{\mathfrak{p}}(a_i(x))=\infty,\\ \operatorname{red}\left(\frac{a_i(x)}{\pi^{v_{\mathfrak{p}}(a_i(x))}}\right) \text{ if } \big(i,v_{\mathfrak{p}}(a_i(x))\big) \text{ lies on } S. \end{array} \right.\] Finally, the residual polynomial of the side \(S\) is the polynomial \[R_S(y)=c_s+c_{s+e}y+\cdots +c_{s+(d-1)e}y^{d-1}+c_{s+de}y^d\in k_{{\mathfrak{p}},\phi}[y].\] Notice that \(c_s\) and \(c_{s+de}\) are always nonzero since they are the initial and terminal vertices, respectively, of the side \(S\). In this work, we will almost always be developing \(f(x)\) with respect to a linear polynomial, so \(k_{{\mathfrak{p}},\phi}=k_{\mathfrak{p}}\), and we will often write the latter to ease notation.

Having established notation, we state a theorem that connects prime splitting and polynomial factorization. The “three dissections" that we will outline below are due to Ore, and the full Montes algorithm is an extension of this. Our statement loosely follows Theorem 1.7 of [30].

Though Theorem 3 describes general \({\mathfrak{p}}\)-adic factorizations, we will be working with monic integral polynomials, so our factorizations will be over the \({\mathfrak{p}}\)-adic integers.

In addition to Ore’s Theorem, it will be convenient to have the following classical result on the irreducibility of radical polynomials. For a reference see [32] or Chapter 6, §9 of [33].

3 Proof of the Main Theorem↩︎

This section is devoted to the proof of Theorem 1. We will proceed through the cases in numerical order. In contrast to the statement of Theorem 1, in this section the choice of \(i\) or \(j\) for an index does not indicate whether a factorization is occurring over \(\mathbb{F}_2\) or \(\mathbb{F}_4\). Likewise, we will freely conflate a polynomial in \(\mathbb{Z}[x]\) with its image in \(\mathbb{F}_p[x]\), \(\mathbb{Q}_p[x]\), and extensions thereof. Throughout we readily use the correspondence between \(p\)-adic factors of an irreducible polynomial over a number field and prime ideal factors of \(p\) in the associated extension. See Proposition II.8.2 of [3]. Finally, to avoid being overly verbose, we often speak of the factorization of \({\mathfrak{p}}\) in \(K\) when we mean the prime ideal factorization of the ideal generated by \({\mathfrak{p}}\) in the ring of integers of \(K\).

First, Case [MainI] is simply Dedekind–Kummer factorization. See [34] or Proposition I.8.3 of [3].

Case [MainII] is covered by Theorem 6.1 of [1]. The proof is an application of Theorem 3. For Cases [MainIII] and [MainIV] we use [7]. We state the result here for convenience and to unify notation. Note that we have fixed a minor typo in Vélez’s Case D) ii) c) (Case [VelezEvenIIIi] for us), and we have included the factorization of \(x^{2^m}-a\) over \(\mathbb{Z}_2[x]\) for reference. Further, we have the theorem for an arbitrary even \(a\) as opposed to an \(a\) such that \(v_2(a)=2^k\). This is justified in Remark 6.

Returning to the proof of Theorem 1, Theorem 5 gives the factorization of \(2\) in the ring of integers of \(\mathbb{Q}\big(\sqrt[2^m]{a}\big)\). Recall \(n=2^m n_0\) with \(n_0\) odd. Our strategy is to describe the factorization of each prime of \({\mathcal{O}}_{\mathbb{Q}(\sqrt[2^m]{a})}\) above 2 in the ring of integers of \(\mathbb{Q}\big(\sqrt[2^m]{a},\sqrt[n_0]{a}\big)=\mathbb{Q}\big(\sqrt[n]{a}\big)\); this is accomplished by Lemma 1 for Case [MainIII] and Lemma 2 for Case [MainIV]. The following diagram summarizes this:

In the end, Figure 1 shows \[2{\mathcal{O}}_{\mathbb{Q}(\sqrt[n]{a})} = \prod\limits_{i=1}^{r}\left(\prod\limits_{j=1}^{s_i}{\mathfrak{P}}_{i,j}^{h_{i,j}}\right)^{e_i}.\]

Applying Lemma 1 to Cases [VelezOddi], [VelezOddii], and [VelezOddiii] of Theorem 5, we have Case [MainIII] of Theorem 1. For Case [MainIV], Dedekind–Kummer factorization is not enough, and we must employ Theorem 3. Note Theorem 5 Cases [VelezEvenI], [VelezEvenII], and [VelezEvenIII] (or Cases [VelezOddi], [VelezOddii], and [VelezOddiii] if \(k=v_2\big(v_2(a)\big)\geq v_2(n)=m\)) describe the splitting of 2 in the ring of integers of \(\mathbb{Q}\big(\sqrt[2^m]{a}\big)\).

We apply Lemma 2 to each of the subcases of Cases [VelezEvenI], [VelezEvenII], and [VelezEvenIII] of Theorem 5, and we obtain Cases [MainIVi], [MainIVii], and [MainIViii] of Theorem 1. If \(k=v_2\big(v_2(a)\big)\geq v_2(n)=m\), then \(\mathbb{Q}\big(\sqrt[2^m]{a}\big)=\mathbb{Q}\big((a_02^{2^kh})^\frac{1}{2^m}\big)=\mathbb{Q}\big(\sqrt[2^m]{a_0}\big)\), so we employ Cases [VelezOddi], [VelezOddii], and [VelezOddiii] with \(x^{2^m}-a_0\) to obtain the splitting of 2 in \(\mathbb{Q}\big(\sqrt[2^m]{a}\big)\). We then apply Lemma 2 to obtain Theorem 1 Case [MainIV0]. This completes our proof of Theorem 1.

3.1 Auxiliary Lemmas and Remarks↩︎

For Theorem 5, it is important to know the largest non-negative integer \(t\) such that \(a_0\), the odd part of \(a\), is in \(\mathbb{Z}_2^{2^t}\) (equivalently, \(\mathbb{Q}_2^{2^t}\)). Asking whether \(a_0\) is a \(2^t\)-th power in \(\mathbb{Z}_2\) is the same as asking whether \(x^{2^t}-a_0\) has a root in \(\mathbb{Z}_2\). We employ Theorem 3. Reducing \(x^{2^m}-a_0\) modulo 2, we obtain \(x^{2^m}-a_0\equiv \big(x-a_0\big)^{2^m}\equiv (x-1)^{2^m}\bmod 2\). Thus we take the \((x-1)\)-adic development: \[\label{Eq:32binomexpansion2} \begin{align} x^{2^m}-a_0&=\left(x-1+1\right)^{2^m}-a_0\\ &=\left(\sum\limits_{i=0}^{2^m}\binom{2^m}{i}\left(x-1\right)^i 1^{2^m-i}\right) - a_0\\ &=\left(\sum\limits_{i=1}^{2^m}\binom{2^m}{i} \left(x-1\right)^i\right) + 1 - a_0. \end{align}\tag{3}\]

Let \(w_0=v_2(a_0-1)\) and let \(S\) denote the left-most side of the principal \((x-1)\)-polygon. The side \(S\) identifies whether we have a root in \(\mathbb{Z}_2\). The initial vertex is \((0,w_0)\). If \(w_0\leq m\), then \(S\) has a fractional slope and does not correspond to a root. If \(w_0>m+1\), then \(S\) has length 1 and we have a root. If \(w_0=m+1\), then \(S\) has length 2 and contains the integer lattice point \((1,m)\). More analysis is needed in this case. These three cases are illustrated in Figure 2.

If \(w_0=m+1\), then \((2,m-1)\) is the terminal vertex of \(S\). We analyze the residual polynomial in \(\mathbb{F}_2[y]\), \[\begin{align} R_S(y)&=\frac{1-a_0}{2^{w_0}}+\frac{2^m}{2^m}y+\frac{\binom{2^m}{2}}{2^{m-1}}y^2\\ &=y^2+y+1. \end{align}\]

We see that the factor of \(x^{2^m}-a_0\) corresponding to the side \(S\) yields a degree 2 extension of \(\mathbb{F}_2\) and not a root in \(\mathbb{Z}_2\). Thus \(x^{2^m}-a_0\) does not have any roots in \(\mathbb{Z}_2\) and \(a_0\notin \mathbb{Z}_2^{2^m}\). (Equivalently, \(a_0\notin \mathbb{Q}_2^{2^m}\).) We summarize with the following lemma.

4 Common \(N\)-Index Divisors↩︎

With this section we will prove Corollaries 1 and 2 and then construct a number of examples. For our proof, we will combine the main theorem of [5] with Theorems 1 and 2.

Recall that a common \(N\)-index divisor (C\(N\)-ID) is a prime that divides the index of any order with \(N\) ring generators in the maximal order. Also, recall that the number of monic irreducible polynomials of degree \(f\) over \(\mathbb{F}_{q}\) is \[\operatorname{Irred}(f,q) = \frac{1}{f}\sum_{d\mid f}\mu\left(\frac{f}{d}\right)q^d.\] Finally, given an integral prime \(p\) and a number field \(K/\mathbb{Q}\), let \(\lambda_f\) be the number of primes of residue class degree \(f\) above \(p\) in \({\mathcal{O}}_K\). Pleasants’s main theorem in the case where the base field is \(\mathbb{Q}\) follows:

Now we proceed with the proof of Corollary 1.

We now undertake the proof of Corollary 2.

Notice that in Example 1 the degree is quite large. From Theorem 7, the minimum degree required for \(p\) to be a common \(N\)-index divisor is \(p^N+1\). We can achieve this via the following construction.

5 Examples of the Main Theorem↩︎

We will illustrate Theorem 1 via a series of examples, varying \(n\) and \(a\) to illuminate different cases. Of course, Case [MainI] is simply Dedekind–Kummer factorization. For example, take \(n=a=3\), though \(a\) could be any odd integer. We see \[x^3-3\equiv x^3-1\equiv (x-1)(x^2+x+1) \bmod 2.\text{ Thus } 2{\mathcal{O}}_{\mathbb{Q}(\sqrt[3]{3})}=\big(2,\sqrt[3]{3}-1\big)\big(2, \sqrt[3]{9}+\sqrt[3]{3}+1\big).\]

6 Non-Monogenic Fields with No Common Index Divisors↩︎

Following Pleasants, call a number field \(K\) exceptional³ if it has no common index divisors yet it is still non-monogenic. When \(n+1\) is prime, Pleasants [5] uses congruence conditions on discriminants to construct radical (pure) number fields of degree \(n\) that have no common index divisors yet are non-monogenic. We will employ the index form to construct other exceptional radical extensions. This construction yields exceptional examples of any degree \(n>2\).

References↩︎