Pólya-Ostrowski Groups and Unit Indices in Real Biquadratic Fields

1. Notation

For a number field K, we denote by $d_K$, $Cl\left(K\right)$, $h_K$, ${\mathcal{O}}_K$ and $U_K$ the discriminant of K, the ideal class group, the class number, the ring of integers and the group of units of K, respectively. For a prime number p, we denote by $e_p(K/\mathbb{Q})$ the ramification index of p in $K/\mathbb{Q}$. The notation $N_{K/\mathbb{Q}}$ denotes both the element norm and the ideal norm morphisms from K to $\mathbb{Q}$.

2. Introduction

The study of integer-valued polynomials on rings of integers of number fields originates in the seminal works of Pólya [8] and Ostrowski [7]. For a number field K with ring of integers ${\mathcal{O}}_K$, let

$$\mathrm{Int}\left({\mathcal{O}}_K\right)=\{f\in K\left[X\right]:f\left({\mathcal{O}}_K\right)\subseteq {\mathcal{O}}_K\}$$

denote the ring of integer-valued polynomials on ${\mathcal{O}}_K$. It is known that $\mathrm{Int}\left({\mathcal{O}}_K\right)$ is a free ${\mathcal{O}}_K$-module [Section 2] [10]. Following Zantema [10], the field K is called a Pólya field if $\mathrm{Int}\left({\mathcal{O}}_K\right)$ admits a regular basis, that is, a basis containing exactly one polynomial of each degree. An equivalent arithmetic formulation is given in terms of the Ostrowski ideals. For each positive integer q, define

$${\Pi }_q\left(K\right)=\prod _{\begin{array}{c}\mathfrak{m}\in \mathrm{Max}\left({\mathcal{O}}_K\right)\\ N_{K/\mathbb{Q}}\left(\mathfrak{m}\right)=q\end{array}}\mathfrak{m},$$

with the convention that ${\Pi }_q\left(K\right)={\mathcal{O}}_K$ if no such ideals exist. The field K is Pólya if and only if all ideals ${\Pi }_q\left(K\right)$ are principal.

Cahen and Chabert [2] introduced the Pólya–Ostrowski group (or simply the Pólya group) $\mathrm{Po}\left(K\right)$ as the subgroup of the ideal class group $\mathrm{Cl}\left(K\right)$ generated by the classes of all Ostrowski ideals. Thus K is Pólya if and only if $\mathrm{Po}\left(K\right)$ is trivial.

When $K/\mathbb{Q}$ is Galois, the Pólya group admits a cohomological description. In this case, the Ostrowski ideals generate precisely the subgroup of strongly ambiguous ideals, and $\mathrm{Po}\left(K\right)$ coincides with the subgroup of $\mathrm{Cl}\left(K\right)$ consisting of strongly ambiguous ideal classes. The reader is referred to [3,5,6] for some results on Pólya fields and Pólya groups.

Zantema [10] established an exact sequence relating $\mathrm{Po}\left(K\right)$ to the cohomology group $H^1(\mathrm{Gal}(K/\mathbb{Q}),U_K)$ and to the ramification indices of primes in K.

Proposition 1 

(Proposition 3.1). [10] For a Galois extension $K/\mathbb{Q}$, the following sequence is exact:

$$0⟶H^1(Gal(K/\mathbb{Q}),U_K)⟶\underset{\mathrm{p}\mathrm{prime}}{⨁}\mathbb{Z}/e_p(K/\mathbb{Q})\mathbb{Z}⟶Po\left(K\right)⟶0,$$

(1)

where $e_p(K/\mathbb{Q})$ denotes the ramification index of a prime p in K.

The above exact sequence allows one to derive restrictions on ramification in Pólya fields. In particular, Zantema [Section 5] [10] showed that in a Pólya real biquadratic field at most five primes ramify, and that this bound is sharp. For instance $K=\mathbb{Q}(\sqrt{35},\sqrt{381})$ is Pólya with five ramifications [page 20] [10] (Some infinite families of Pólya real biquadratic fields with maximum ramification have been found in [6]). For a real biquadratic field K, Bennett Setzer [9] described the structure of $H^1(Gal(K/\mathbb{Q}),U_K)$ in terms of unit-theoretic data from the quadratic subfields $k_1,k_2,k_3$, a description that has been used in several subsequent works, including earlier paper of the second author [6] The present paper builds on these foundational results but differs in scope and outcome: rather than focusing on particular families of fields, we obtain an explicit and uniform formula for the order of the Pólya group of an arbitrary real biquadratic fields K in terms of the Hasse unit index $\left[U_K:U_{k_1}{U}_{k_2}{U}_{k_3}\right]$, ramification indices, and the number of quadratic subfields whose fundamental units have negative norm, see Theorem 1. This leads, as an application, to refined upper bounds on the number of ramified primes in Pólya real biquadratic fields, improving Zantema’s classical bound in several cases, see Corollary 1.

3. An Explicit Relation Between the Order of the Pólya Groups and the Hasse Unit Indices of Real Biqauadratic Fields

In this section, summarizing some results of Bennett Setzer [9] for real biqudratic fields K, we find an explicit relation between $\#Po\left(K\right)$ and the Hasse unit index of K.

Proposition 2. 

[9] Let $K=\mathbb{Q}(\sqrt{d_1},\sqrt{d_2})$ be a real biquadratic field with Galois group G, where $d_1,d_2\ne 1$ are two different square-free positive integers. Denote by $k_1,k_2,k_3$ the three real quadratic subfields of K. Let $H=H^1(G,U_K)$, and $H\left[2\right]$ be the 2-torsion subgroup of H. Then, the following assertions hold.

(i)

[9] $\left[H:H\left[2\right]\right]\le 2$. Moreover $\left[H:H\left[2\right]\right]=2$ if and only if 2 ramifies totally in $K/\mathbb{Q}$ and all $k_i$’s contain elements with the same norm either 2 or $-2$.

(ii)

[9] Denote by t the number of quadratic subfields of K whose fundamental units have negative norm. Then

$$\left[U_K:U_{k_1}{U}_{k_2}{U}_{k_3}\right].\#H\left[2\right]=\left\{\begin{array}{cc}{2}^{5-t}\hfill & :t=0,1,\hfill \\ 2^3\hfill & :t=2,3.\hfill \end{array}\right.$$

Moreover for $t=3$, $H\left[2\right]$ contains a subgroup of order 4.

(iii)

For $i=1,2,3$, let ${\alpha }_i$ be the generator of units in $k_i$ with positive norms. If at least one $k_i$ has no units of negative norm, then $H\left[2\right]$ is isomorphic to the subgroup

$$\mathcal{A}=\langle \left[d_1\right],\left[d_2\right],\left[N_{k_1/\mathbb{Q}}({\alpha }_1+1)\right],\left[N_{k_2/\mathbb{Q}}({\alpha }_2+1)\right],\left[N_{k_1/\mathbb{Q}}({\alpha }_3+1)\right]\rangle$$

(2)

of the quotient group ${\mathbb{Q}}^{\times }/{\mathbb{Q}}^{\times 2}$. Otherwise, $H\left[2\right]$ would be isomorphic to the subgroup $\mathcal{A}$ with possibly one more generator.

For a real biquadratic field K, the first cohomology group $H^1(Gal(K/\mathbb{Q}),U_K)$ can be seen as a “common term” in the above results of Bennet Setzer and the exact sequence (1). This observation leads us to relate the order of Pólya groups of real biquadratic fields to their Hasse unit indices.

Theorem 1. 

Let $K=\mathbb{Q}(\sqrt{d_1},\sqrt{d_2})$ be a real biquadratic field with quadratic subfields $k_i=\mathbb{Q}\left(\sqrt{d_i}\right)$ ($d_3$ is the squarefree part of $d_1{d}_2$). Denote by t the number of quadratic subfields of K whose fundamental unit have negative norm.

(i)

If 2 ramifies totally in $K/\mathbb{Q}$ and all $k_i$’s contain elements with the same norm either 2 or $-2$, then:

$$\#Po\left(K\right)=\left\{\begin{array}{cc}{\displaystyle \frac{\left[U_K:U_{k_1}{U}_{k_2}{U}_{k_3}\right].{\prod }_{p\mid d_K}{e}_p(K/\mathbb{Q})}{2^{6-t}}}\hfill & :t=0,1,\hfill \\ \hfill & \\ {\displaystyle \frac{\left[U_K:U_{k_1}{U}_{k_2}{U}_{k_3}\right].{\prod }_{p\mid d_K}{e}_p(K/\mathbb{Q})}{2^4}}\hfill & :t=2,3.\hfill \end{array}\right.$$

(ii)

Otherwise:

$$\#Po\left(K\right)=\left\{\begin{array}{cc}{\displaystyle \frac{\left[U_K:U_{k_1}{U}_{k_2}{U}_{k_3}\right].{\prod }_{p\mid d_K}{e}_p(K/\mathbb{Q})}{2^{5-t}}}\hfill & :t=0,1,\hfill \\ \hfill & \\ {\displaystyle \frac{\left[U_K:U_{k_1}{U}_{k_2}{U}_{k_3}\right].{\prod }_{p\mid d_K}{e}_p(K/\mathbb{Q})}{2^3}}\hfill & :t=2,3.\hfill \end{array}\right.$$

In particular, for $t\in \{2,3\}$ Pólya groups have the same order.

Proof. 

We only prove part (i), and the second part is proved similarly. Suppose that 2 ramifies totally in $K/\mathbb{Q}$, and all $k_i$’s contain elements with same norm either 2 or $-2$. Then, by the first two parts of Proposition 2, we have

$$\#H^1(Gal(K/\mathbb{Q}),U_K)·\left[U_K:U_{k_1}{U}_{k_2}{U}_{k_3}\right]=\left\{\begin{array}{cc}{2}^{6-t}\hfill & :t=0,1,\hfill \\ 2^4\hfill & :t=2,3.\hfill \end{array}\right.$$

(3)

On the other hand, by the exact sequence (1),

$$\#Po\left(K\right)={\displaystyle \frac{{\prod }_{p\mid d_K}{e}_p(K/\mathbb{Q})}{\#H^1(Gal(K/\mathbb{Q}),U_K)}}.$$

(4)

Using (3) with (4), we get the assertion.

□

For a real biquadratic field K, we can use Proposition (2) to find a lower bound for the order of the first cohomology group $H^1(Gal(K/\mathbb{Q}),U_K)$. Then using the exact sequence (1) we can compute (or at least approximate) the order of $Po\left(K\right)$, and then Theorem (1) gives us the corresponding Hasse unit index of K. To do so, we also need the following property of a generator of positive norm units in real quadratic fields.

Proposition 3. 

[6] Let $k=\mathbb{Q}\left(\sqrt{d}\right)$ be a real quadratic field and ${\alpha }_k>1$ be the generator of positive norm units in k. Assume that $n_k$ is the squarefree part of $N_{k/\mathbb{Q}}({\alpha }_k+1)$. Then $n_k\notin \{1,d\}$, and $n_k$ divides the discriminant of k.

Example 1. 

Let $p\equiv q\equiv 1\left(\mathrm{mod}4\right)$ be distinct prime numbers, $K=\mathbb{Q}(\sqrt{p},\sqrt{q})$, $k_1=\mathbb{Q}\left(\sqrt{p}\right)$, $k_2=\mathbb{Q}\left(\sqrt{q}\right)$, $k_3=\mathbb{Q}\left(\sqrt{pq}\right)$ and denote the fundamental unit of $k_i$ by ${ϵ}_i$, respectively. Then $N_{k_1/\mathbb{Q}}\left({ϵ}_1\right)=N_{k_2/\mathbb{Q}}\left({ϵ}_2\right)=-1$, see e.g. [4]. By Theorem (1), we get

$$2.\#Po\left(K\right)=\left[U_K:U_{k_1}{U}_{k_2}{U}_{k_3}\right].$$

(5)

Let $H=H^1(Gal(K/\mathbb{Q}),U_K)$. Since 2 doesn’t ramify totally in $K/\mathbb{Q}$, part $\left(i\right)$ in Proposition (2) implies that $H=H\left[2\right]$. Using Propositions 1, 2 and 3, we have

•

if $N_{k_3/\mathbb{Q}}\left({ϵ}_3\right)=+1$, then

$$〈p,q〉\left(\mathrm{mod}{\mathbb{Q}}^{\times 2}\right)\subseteq \mathcal{A}\simeq H\left[2\right]=H\subseteq {\displaystyle \frac{\mathbb{Z}}{2\mathbb{Z}}}\oplus {\displaystyle \frac{\mathbb{Z}}{2\mathbb{Z}}};$$

•

if $N_{k_3/\mathbb{Q}}\left({ϵ}_3\right)=-1$, then

$${\displaystyle \frac{\mathbb{Z}}{2\mathbb{Z}}}\oplus {\displaystyle \frac{\mathbb{Z}}{2\mathbb{Z}}}\subseteq H\left[2\right]=H\subseteq {\displaystyle \frac{\mathbb{Z}}{2\mathbb{Z}}}\oplus {\displaystyle \frac{\mathbb{Z}}{2\mathbb{Z}}}.$$

Hence, in both cases, $H^1(Gal(K/\mathbb{Q}),U_K)\simeq {\displaystyle \frac{\mathbb{Z}}{2\mathbb{Z}}}\oplus {\displaystyle \frac{\mathbb{Z}}{2\mathbb{Z}}}$ and by the exact sequence (1), K is a Pólya field. By the relation (5), we find $\left[U_K:U_{k_1}{U}_{k_2}{U}_{k_3}\right]=2$.

Remark 1. 

Note that in the above example $k_1$ and $k_2$ are Pólya. Indeed, in almost cases, compositum of two quadratic Pólya fields is a Pólya field [5] for which unit index is obtained immediately from Theorem (1).

As mentioned before, Zantema [10] gave a sharp upper bound on the number of ramified primes in Pólya real biquadratic fields.

Proposition 4 

(Section 5). [10] Let K be a real biquadratic field, and denote by $s_K$ the number of primes that are ramified in $K/\mathbb{Q}$. If K is Pólya, then $s_K\le 5$. Moreover, this upper bound is sharp.

As a direct consequence of Theorem 1, we obtain a refined version of Zantem’s upper bound in the above proposition.

Corollary 1. 

Let K be a real biquadratic field. Denot by t the number of quadratic subfields of K whose fundamental units have negative norm, and by $s_K$ the number of primes that are ramified in $K/\mathbb{Q}$. If K is Pólya, then

$$s_K\le \left\{\begin{array}{cc}\hfill 5:& t=0,\hfill \\ \hfill 4:& t=1,\hfill \\ \hfill 3:& t=2,3.\hfill \end{array}\right.$$

Proof. 

Suppose that $Po\left(K\right)=0$. We prove the cases $t=2,3$. The other two cases, say $t=0,1$, can be proved similarly. Let $k_1,k_2,k_3$ be the three quadratic subfield of K, and suppose that $k_1$ and $k_2$ have some units of negative norm. If 2 is totally ramified in $K/\mathbb{Q}$, and all $k_i$’s contain elements with the same norm either 2 or $-2$, then by part (i) of Theorem 1, we get

$$\prod _{p|d_K}{e}_p(K/\mathbb{Q})=e_2(K/\mathbb{Q})·\prod _{\begin{array}{c}p|d_K\\ p\ne 2\end{array}}{e}_p(K/\mathbb{Q})={\displaystyle \frac{2^4}{\left[U_K:U_{k_1}{U}_{k_2}{U}_{k_3}\right]}},$$

which implies that

$$\prod _{\begin{array}{c}p|d_K\\ p\ne 2\end{array}}{e}_p(K/\mathbb{Q})\le 2^2,$$

or equivalently, $s_K\le 3$. On the other hand, if 2 doesn’t ramify totally in $K/\mathbb{Q}$ or one of the equations

$$N_{k_i/\mathbb{Q}}\left(x_i\right)=+2\mathrm{or}-2,i=1,2,3,$$

doesn’t have any solution $x_i\in {\mathcal{O}}_{k_i}$, then by part (ii) of Theorem 1, we obtain

$$\prod _{p|d_K}{e}_p(K/\mathbb{Q})={\displaystyle \frac{2^3}{\left[U_K:U_{k_1}{U}_{k_2}{U}_{k_3}\right]}}\le 2^3,$$

and again we get $s_K\le 3$. □