1 Introduction

Arithmetic properties of solutions of norm form equations have a considerable literature. Here we mention only one result (which is the most important from our viewpoint), and for generalities on norm form equations or other related result, we refer the interested reader to the papers [1] or [2]. Fuchs and Heintze considered terms of so-called multi-recurrences (being natural generalizations of linear recurrence sequences) among the coordinates of solutions of norm form equations. They set the problem in the general case, however, their results concern the case of multi-recurrences which are simple. Simplifying their result to the case of linear recurrence sequences, this means that the characteristic polynomials of these sequences have simple zeroes. They could prove that a simple multi-recurrence, under certain necessary assumptions, can have only finitely many terms among the coordinates of solutions of a norm form equation.

In the special situation when the underlying field is quadratic, we in fact investigate terms of recurrence sequences among the solutions of generalized Pell equations. For an account on these results, see [4]. In particular, the main theorem of [4] states that under certain assumptions, a recurrence sequence has only finitely many terms among the coordinates of solutions of generalized Pell equations.

In this paper, we provide a common extension of the main results of [2] (more precisely, its corollary to the case of linear recurrence sequences) and of [4]. That is, we prove that, under certain assumptions, an arbitrary linear recurrence sequence can have only finitely many terms in the coordinates of solutions of norm form equations. We show that the imposed assumptions are necessary. In our proof, we shall combine the finiteness of solutions of polynomial-exponential equations proved by Schlickewei and Schmidt [5] with some other tools and ideas.

2 The main result

To formulate our main result, we need to introduce some notation. Let K be an algebraic number field of degree k, and write \({\mathcal {N}}(\alpha )\) for the norm of \(\alpha \in K\) (over \(\mathbb {Q}\)). Let \(\alpha _1,\ldots ,\alpha _k \in K\) be linearly independent over \(\mathbb {Q}\), and let m be a non-zero integer. Consider the norm form equation

$$\begin{aligned} {\mathcal {N}}(x_1\alpha _1+\cdots +x_k\alpha _k)=m \end{aligned}$$
(1)

in integers \(x_1,\ldots ,x_k\). Write \(X_i\) \((i=1,\ldots ,k)\) for the coordinate sets of solutions of (1).

Let r be a positive integer, \(a_1,\dots ,a_r\in \mathbb {Z}\) such that \(a_r\ne 0\), and \(U_0,\dots ,U_{r-1}\in \mathbb {Z}\) not all zero. If

$$\begin{aligned} U_n = a_1U_{n-1} + \cdots +a_rU_{n-r}\ \ \ (n\ge r) \end{aligned}$$
(2)

and r is minimal such that \((U_n)\) satisfies a relation above, then \(U=(U_n)=(U_n)_{n\ge 0}\) is called a linear recurrence sequence (of integers) of order r. Throughout this paper, we always assume that a recurrence sequence is given by its minimal length relation (2). We shall also use the notation

$$\begin{aligned} U=U(a_1,\dots ,a_r,U_0,\ldots ,U_{r-1}). \end{aligned}$$

The characteristic polynomial of \((U_n)\) is defined by

$$\begin{aligned} f(x):= x^r - a_1x^{r-1}-\cdots -a_r = \prod _{i=1}^d (x - \beta _i)^{m_i} \end{aligned}$$
(3)

where \(\beta _1,\ldots ,\beta _d\) are distinct algebraic numbers and \(m_1,\dots ,m_t\) are positive integers. Then as it is well-known (see e.g. [9, Theorem C.1 in part C]), we have

$$\begin{aligned} U_n=\sum _{i=1}^d g_i(n)\beta _{i}^n \ \ \ (n\ge 0). \end{aligned}$$
(4)

Here \(g_i(x)\) is a not identically zero polynomial of degree at most \(m_i-1\) \((i=1,\dots ,s)\) with coefficients in the number field \(\mathbb {Q}(\beta _1,\ldots ,\beta _d)\). (The fact that the polynomials \(g_i\) are not identically zero is not formulated in [9, Theorem C.1], however it is clear from its proof.) We say that the sequence \((U_n)\) is degenerate if there are integers i, j with \(1\le i<j\le s\) such that \(\alpha _i/\alpha _j\) is a root of unity; otherwise it is non-degenerate.

Now we can formulate our main result.

Theorem 2.1

Let K be an algebraic number field of degree k, and \((U_n)\) be a non-degenerate linear recurrence sequence of integers of order \(r\ge 2\) given by (2) and (4). If \(\beta _i\ne \pm 1\) for some \(i=1,\dots ,d\) and one of the conditions

  1. (i)

    \(a_r\ne \pm 1\),

  2. (ii)

    \(K\cap \mathbb {Q}(\beta _i)=\mathbb {Q}\ (i=1,\ldots ,d)\)

is satisfied, then

$$\begin{aligned} U_n\in X_1 \cup \cdots \cup X_k \end{aligned}$$
(5)

holds only for finitely many indices n, where \(X_i\) \((i=1,\dots ,k)\) are the sets of the coordinates of the solutions of any norm form equation, as defined in (1). Further, the number of such indices is bounded by \(c_1\), where \(c_1=c_1(m,k,r)\) is an effectively computable constant depending only on m, k, r.

Remark

It is important to mention that (1) can have infinitely many solutions (with m chosen appropriately), unless K is \(\mathbb {Q}\) or an imaginary quadratic field. This follows from results of Schmidt (see [6, Satz 2] or [7, Chapter VII]). In certain cases, (1) can have infinitely many solutions even if we take only \(k'\) linearly independent algebraic integers \(\alpha _i\in K\) \((i=1,\dots ,k')\) with \(k'<k\). This happens e.g. if we take \(\alpha _1=1\) and \(\alpha _2=\sqrt{2}\) in \(K=\mathbb {Q}(\root 4 \of {2})\), together with \(m=1\). (For a precise description of these cases, see [6, 7] again). Clearly, Theorem 2.1 remains valid also in these cases. Indeed, choose algebraic integers \(\alpha _{k'+1},\dots ,\alpha _k\) from K such that \(\alpha _1,\ldots ,\alpha _k\) are linearly independent over \(\mathbb {Q}\). As \(X_1\cup \cdots \cup X_{k'}\subseteq X_1\cup \cdots \cup X_k\), the finiteness of the set of indices n with

$$\begin{aligned} U_n\in X_1 \cup \cdots \cup X_{k'} \end{aligned}$$

immediately follows from Theorem 2.1.

We also note that the conditions in the theorem are all necessary. To show this, we exhibit some examples. However, we do so after the proof of Theorem 2.1 since then we shall have the required machinery and notation.

3 Lemmas and proofs

To prove our theorem, we need two lemmas. The first one is due to Schlickewei and Schmidt [5]. It concerns the finiteness of the solutions of polynomial-exponential equations. For its formulation, we need to introduce some new notation.

Let L be an algebraic number field, and let \(P_1,\dots ,P_s\) be not identically zero polynomials in t variables over L. Further, let \(\kappa _{i1},\dots ,\kappa _{it}\) \((i=1,\dots ,s)\) be non-zero elements of L. Consider the equation

$$\begin{aligned} \sum _{i=1}^s P_i(\varvec{x})\varvec{\kappa }_i^{\varvec{x}}=0 \end{aligned}$$
(6)

in tuples \(\varvec{x}=(x_1,\dots ,x_t)\in \mathbb {Z}^t\), where \(\varvec{\kappa }_i=(\kappa _{i1},\ldots ,\kappa _{it})\) \((i=1,\ldots ,s)\) and

$$\begin{aligned} \varvec{\kappa }_i^{\varvec{x}}=\kappa _{i1}^{x_1}\cdots \kappa _{it}^{x_t}\ \ \ (i=1,\ldots ,s). \end{aligned}$$
(7)

Let \({{\mathcal {P}}}\) be a partition of the set \(\Lambda =\{1,\ldots ,s\}\). Then the system of equations

$$\begin{aligned} \sum _{i \in \lambda } P_i(\varvec{x})\varvec{\kappa }_i^{\varvec{x}}=0\ \ \ (\lambda \in {{\mathcal {P}}}) \end{aligned}$$
(8)

is called a refinement of (6). Let \({{\mathcal {S}}}(\mathcal P)\) be the set of solutions of (8) which are not solutions of

$$\begin{aligned} \sum _{i \in \lambda } P_i(\varvec{x})\varvec{\kappa }_i^{\varvec{x}}=0\ \ \ (\lambda \in {{\mathcal {P}}}') \end{aligned}$$

with any proper refinement \({{\mathcal {P}}}'\) of \({{\mathcal {P}}}\). Set \(i_1\overset{{\mathcal {P}}}{\sim }i_2\) if \(i_1\) and \(i_2\) lie in the same subset \(\lambda \) of \({{\mathcal {P}}}\). Let \(G({{\mathcal {P}}})\) be the subgroup of \(\mathbb {Z}^t\) consisting of tuples \(\varvec{z}=(z_1,\dots ,z_t)\) with

$$\begin{aligned} \varvec{\kappa }_{i_1}^{\varvec{z}}=\varvec{\kappa }_{i_2}^{\varvec{z}}\ \ \ \text {for any}\ i_1,i_2\ \text {with}\ i_1\overset{\mathcal P}{\sim }i_2. \end{aligned}$$

Lemma 3.1

Using the above notation, if \(G(\mathcal {P})=\left\{ \varvec{0}\right\} \), then we have

$$\begin{aligned} \left| \mathcal {S}(\mathcal {P})\right| < 2^{35A^3} D^{6A^2} \end{aligned}$$

with \(D=\deg (L)\) and

$$\begin{aligned} A=\max \left( t,\sum \limits _{i\in \Lambda } \left( {\begin{array}{c}t+\delta _i\\ t\end{array}}\right) \right) , \end{aligned}$$

where \(\delta _i\) is the total degree of the polynomial \(P_i\).

Proof

The statement is [5, Theorem 1]. \(\square \)

Our second lemma is an application of Lemma 3.1 with \(t=1\) to linear recurrence sequences. Note that it is closely related to the zero multiplicity of linear recurrence sequences, known to be finite and bounded due to a deep result of Schmidt [8].

Lemma 3.2

Let \((U_n)\) be a non-degenerate linear recurrence sequence of integers of order \(r\ge 2\). Using (4), let \(I \subseteq \{1,\ldots ,d\}\). Then there are only finitely many indices n for which

$$\begin{aligned} \sum _{i \in I} g_i(n)\beta _{i}^n=0 \end{aligned}$$
(9)

holds. Further, the number of such indices n can be bounded by \(c_2\), where \(c_2=c_2(r)\) is an effectively computable constant depending only on r.

Proof

We apply Lemma 3.1 with \(t=1\). Let n be a solution of (9). First observe that if \(g_i(n)=0\) for some \(i\in I\), then n comes from a finite set of cardinality bounded in terms of r. So we may assume that \(g_i(n)\ne 0\) \((i\in I)\). Clearly, there is a partition \({\mathcal {P}}\) of I such that \(n\in {{\mathcal {S}}}({{\mathcal {P}}})\). If I has a class \(\lambda \) with \(|\lambda |=1\), then n is a root of one of the polynomials \(g_i\) \((i\in I)\), which is excluded. So we can suppose that \(|\lambda |\ge 2\) for all \(\lambda \in {{\mathcal {P}}}\). Clearly, since \((U_n)\) is non-degenerate, we have \(G({{\mathcal {P}}})=\{0\}\). Thus, by Lemma 3.1, n comes again from a finite set of cardinality bounded in terms of r, and our claim follows. \(\square \)

Now we can prove our main result.

Proof of Theorem 2.1

By Lemma 5 of Győry [3], we know that there are only finitely many pairwise non-associate algebraic integers \(\mu \) in K of norm m, and their number can be bounded in terms of k, m. That is, if \((x_1,\dots ,x_k)\in \mathbb {Z}^k\) is any solution of (1), then we have

$$\begin{aligned} x_1\alpha _1+\cdots +x_k\alpha _k=\varepsilon \mu \end{aligned}$$
(10)

with such a \(\mu \), where \(\varepsilon \) is a unit of K. As we need to bound the number of indices n for which \(U_n=x_i\) for some solution \((x_1,\dots ,x_k)\) and i with \(1\le k\), we may assume that \(\mu \) is fixed. Let \(\varepsilon _1,\ldots ,\varepsilon _\ell \) be a system of fundamental units in K. Then (10) yields

$$\begin{aligned} x_1\alpha _1+\cdots +x_k\alpha _k=\delta \varepsilon _1^{u_1}\cdots \varepsilon _\ell ^{u_\ell }\mu , \end{aligned}$$
(11)

where \(\delta \) is a root of unity in K. Since the number of roots of unity in K is bounded in terms of k, we may assume that here \(\delta \) is fixed. Taking the conjugates in (11) (following arguments from [1] and [2]), we get

$$\begin{aligned} \begin{pmatrix} \sigma _1(\alpha _1) &{} \cdots &{} \sigma _1(\alpha _k)\\ \vdots &{} \ddots &{} \vdots \\ \sigma _k(\alpha _1) &{} \cdots &{} \sigma _k(\alpha _k) \end{pmatrix} \begin{pmatrix} x_1\\ \vdots \\ x_k \end{pmatrix} = \begin{pmatrix} \sigma _1(\delta \mu )\nu _{11}^{u_1}\cdots \nu _{1\ell }^{u_\ell }\\ \vdots \\ \sigma _k(\delta \mu )\nu _{k1}^{u_1}\cdots \nu _{k\ell }^{u_\ell } \end{pmatrix}, \end{aligned}$$
(12)

where \(\sigma _1,\ldots ,\sigma _k\) are the isomorphisms of K into \(\mathbb {C}\) (in any order), and \(\nu _{ij}=\sigma _i(\varepsilon _j)\) \((1\le i\le k,\ 1\le j\le \ell )\). As the determinant of the matrix on the left hand side of (12) is known to be non-zero, we can write

$$\begin{aligned} x_i=b_{1i}\nu _{11}^{u_1}\cdots \nu _{1\ell }^{u_\ell }+\cdots +b_{ki}\nu _{k1}^{u_1}\cdots \nu _{k\ell }^{u_\ell }\ \ \ (i=1,\ldots ,k) \end{aligned}$$
(13)

with some algebraic numbers \(b_{ij}\) (belonging to the normal closure of K). Then, using the notation (7), by (4) and (13), relation (5) gives

$$\begin{aligned} b_{1i} \varvec{\nu }_1^{\varvec{u}} + \cdots + b_{ni} \varvec{\nu }_n^{\varvec{u}} = P_1(n)\beta _1^n+\cdots +P_d(n)\beta _d^n \end{aligned}$$
(14)

with \(\varvec{u}=(u_1,\ldots ,u_\ell )\) for some i with \(1\le i\le k\). We may clearly assume that i is fixed. We shall apply Lemma 3.1 to handle the solutions \(\varvec{u},n\) of (14). For this, we need to introduce some new notation. Define the \((k+1)\)-tuples \(\varvec{\vartheta }_j\) \((1\le j\le k+d)\) by

$$\begin{aligned} \varvec{\vartheta }_j= {\left\{ \begin{array}{ll} (\nu _{j1},\ldots ,\nu _{j\ell },1)&{}\text {for}\ 1\le j\le k,\\ (1,\ldots ,1,\beta _{j-k})&{}\text {for}\ k+1\le j\le k+d, \end{array}\right. } \end{aligned}$$

and the polynomials \(Q_j\) \((1\le j\le k+d)\) in \(\ell +1\) variables \({{\varvec{z}}}=(z_1,\ldots ,z_{\ell +1})\) by

$$\begin{aligned} Q_j(\varvec{z})= {\left\{ \begin{array}{ll} b_{ji}&{}\text {for}\ 1\le j\le k,\\ -P_j(z_{k+1})&{}\text {for}\ k+1\le i\le k+d. \end{array}\right. } \end{aligned}$$

Then we can rewrite (14) as

$$\begin{aligned} \sum \limits _{j=1}^{k+d} Q_j(\varvec{z})\varvec{\vartheta }_j^{\varvec{z}}=0. \end{aligned}$$
(15)

Let \({\mathcal {P}}\) be any partition of the set \(J=\{1,\ldots ,k+d\}\). Observe that the number of such partitions is bounded in terms of k and r. Consider the refinement

$$\begin{aligned} \sum \limits _{j\in \lambda } Q_j(\varvec{z})\varvec{\vartheta }_j^{\varvec{z}}=0\ \ \ (\lambda \in {{\mathcal {P}}}) \end{aligned}$$
(16)

of (15). We shall be concerned with the solutions of (16) which are not solutions of proper refinements of it; that is, with \({{\mathcal {S}}}({{\mathcal {P}}})\). If there is a \(\lambda \in {{\mathcal {P}}}\) such that \(\lambda \subseteq \{k+1,\dots ,k+d\}\), then any solution of (16) comes from a case where we have a vanishing subsum in the right hand side of (14). However, since by Lemma 3.2 the number of indices n which allow this is bounded in terms of r, we may assume that it is not the case. We study the set \(G({{\mathcal {P}}})\) - in fact, we show that \(G({\mathcal P})=\{\varvec{0}\}\) in any case. Take an index \(j_1\) with \(k+1\le j_1\le k+d\) such that \(\beta _{j_1}\ne \pm 1\). (By our assumptions, such an index exists.) Then \(j_1\in \lambda \) for some \(\lambda \in {{\mathcal {P}}}\). By the above argument, we see that there is a \(j_2\in \lambda \) with \(1\le j_2\le k\) such that \(Q_{j_2}\) is not identically zero. If \({\varvec{z}}\in G({{\mathcal {P}}})\), then as \(j_1\overset{{\mathcal {P}}}{\sim }j_2\), we have

$$\begin{aligned} \nu _{j_21}^{z_1}\cdots \nu _{j_2\ell }^{z_\ell }=\beta _{j_1}^{z_{\ell +1}}. \end{aligned}$$
(17)

If (i) holds, then since \(\beta _{j_1}\) is not a unit in \(\sigma _{j_2}(K)\), we get \(z_{\ell +1}=0\). Then, as \(\nu _{j_21},\ldots ,\nu _{j_2\ell }\) is a system of fundamental units in \(\sigma _{j_2}(K)\), we obtain \(z_v=0\) \((1\le v\le \ell )\). Hence \(\varvec{z}=\varvec{0}\), so \(G({\mathcal P})=\{\varvec{0}\}\), and our statement follows from Lemma 3.1 in this case. Assume that (ii) holds. Taking the inverse \(\sigma _{j_2}^{-1}\) of \(\sigma _{j_2}\), (17) yields

$$\begin{aligned} \varepsilon _1^{z_1}\cdots \varepsilon _\ell ^{z_\ell }=\beta _h^{z_{\ell +1}}, \end{aligned}$$
(18)

where \(\beta _h=\sigma _{j_2}^{-1}(\beta _{j_1})\). Clearly, \(\beta _h\) is an algebraic conjugate of \(\beta _{j_2}\), so it is a root of f(x) in (3). Thus (ii) gives that the unit \(\beta _h^{z_{\ell +1}}\) is rational - that is, it is \(\pm 1\). Since \(\beta _{j_1}\ne \pm 1\) and by the non-degenerate property of \((U_n)\), \(\beta _h\) cannot be a root of unity, this gives \(z_{\ell +1}=0\). Then as \(\varepsilon _1,\ldots ,\varepsilon _\ell \) form a system of fundamental units in K, we also get \(z_1=\cdots =z_\ell =0\). So \(\varvec{z}=\varvec{0}\), and \(G({{\mathcal {P}}})=\{\varvec{0}\}\). Thus our statement follows from Lemma 3.1 also in this case. The proof of the theorem is complete. \(\square \)

As we mentioned earlier, the assumptions made in Theorem 2.1 are necessary. We conclude our paper by some examples showing that this is the case indeed.

Example

Let \((U_n)\) be given by \(U_0=0\), \(U_1=1\), and \(U_{n+2}=U_{n+1}-U_n\) \((n\ge 0)\). Then as one can easily check, \((U_n)\) is given by \(0,1,1,0,-1,-1,0,1,\ldots \). Taking \(K=\mathbb {Q}(\sqrt{2})\) (so \(k=2\)) and \(\alpha _1=1\), \(\alpha _2=\sqrt{2}\), \(m=1\), (1) reads as

$$\begin{aligned} {\mathcal {N}}(x_1+x_2\sqrt{2})=1 \end{aligned}$$
(19)

in integers \(x_1,x_2\). Since \(1+\sqrt{2}\) is a fundamental unit in K of norm \(-1\), all solutions \(x_1,x_2\) of (19) come from the coefficients of 1 and \(\sqrt{2}\) in \(\pm (1+\sqrt{2})^\ell \) with \(\ell \in \mathbb {Z}\), \(\ell \) even. From this, we easily obtain that

$$\begin{aligned} X_1=\{\pm P_\ell :\ell \ge 0\},\ \ \ X_2=\{\pm Q_\ell :\ell \ge 0\}, \end{aligned}$$

where \(P_0=1\), \(P_1=3\), \(Q_0=0\), \(Q_1=2\), and

$$\begin{aligned} P_{\ell +2}=6P_{\ell +1}-P_\ell ,\ \ \ Q_{\ell +2}=6Q_{\ell +1}-Q_\ell . \end{aligned}$$

In particular, this shows that \(U_n\in \ X_1\cup X_2\) for every \(n\ge 0\). However, in this case, the characteristic polynomial of \((U_n)\) is \(f(x)=x^2-x+1\), so the sequence is degenerate. Hence it is necessary to exclude this property.

Next, consider the sequence \((U_n)\) defined by \(U_0=0\), \(U_1=1\), and \(U_{n+2}=2U_{n+1}-U_n\) \((n\ge 0)\). Then one readily gets that \(U_n=n\) \((n\ge 0)\). So considering again (19), we obtain that \(U_n\in X_1\cup X_2\) for infinitely many n, namely, whenever n belongs to either \((P_\ell )\) or \((Q_\ell )\). Now we have \(f(x)=x^2-2x+1=(x-1)^2\), so the condition that one of the roots of f is different from \(\pm 1\) is violated.

Finally, assume that none of the conditions (i) and (ii) in Theorem 2.1 is satisfied. Let \(\alpha =\root 3 \of {2}\) and \(K=\mathbb {Q}(\alpha )\). Then \(1, \alpha , \alpha ^2\) form an integral basis of K. Further, \(\alpha -1\) is a fundamental unit of K, and the only roots of unity in K are \(\pm 1\). Putting \((\alpha _1,\alpha _2,\alpha _3)=(1,\alpha ,\alpha ^2)\) and \(m=1\), the norm form equation (1) is given by

$$\begin{aligned} {{\mathcal {N}}}(x_1+x_2\alpha +x_3\alpha ^2)=1. \end{aligned}$$

Since the norm of \(\alpha -1\) is 1, this equation is equivalent to

$$\begin{aligned} x_1+x_2\alpha +x_3\alpha ^2=(\alpha -1)^u\ \ \ (u\in \mathbb {Z}). \end{aligned}$$

Thus, in this case, (12) reads as

$$\begin{aligned} \begin{pmatrix} 1 &{} \alpha &{} \alpha ^2\\ 1 &{} \xi \alpha &{} \xi ^2\alpha ^2\\ 1 &{} \xi ^2\alpha &{} \xi \alpha ^2 \end{pmatrix} \begin{pmatrix} x_1\\ x_2\\ x_3 \end{pmatrix} = \begin{pmatrix} (\alpha -1)^u\\ (\xi \alpha -1)^u\\ (\xi ^2\alpha -1)^u \end{pmatrix}, \end{aligned}$$

where \(\xi =(-1+\sqrt{-3})/2\) (which is a primitive third root of unity). From this, by a simple calculation, we obtain that

$$\begin{aligned} x_1=\frac{1}{3}(\alpha -1)^u+\frac{1}{3}(\xi \alpha -1)^u+\frac{1}{3}(\xi ^2\alpha -1)^u\ \ \ (u\in \mathbb {Z}). \end{aligned}$$
(20)

Set \(U_0=1\), \(U_1=-1\), \(U_2=1\), and \(U_{n+3}=-3U_{n+2}-3U_{n+1}+U_n\) \((n\ge 0)\). So \((U_n)\) is a linear recurrence sequence of order \(r=3\), with characteristic polynomial \(f(x)=x^3+3x^2+3x-1\). As one can easily check, the roots of f(x) are given by \(\alpha -1\), \(\xi \alpha -1\), \(\xi ^2\alpha -1\), and also that in this case (4) is given by

$$\begin{aligned} U_n=\frac{1}{3}(\alpha -1)^n+\frac{1}{3}(\xi \alpha -1)^n+\frac{1}{3}(\xi ^2\alpha -1)^n\ \ \ (n\ge 0). \end{aligned}$$
(21)

Comparing (20) and (21), we see that \(U_n\in X_1\) for every \(n\ge 0\). Observe that now (i) does not hold as \(a_3=-1\), and (ii) does not hold since \(K\cap \mathbb {Q}(\alpha -1)=K\). So we need to require the validity of (i) or (ii), indeed.