2. A criterion for completely normal elements

Let $A=\mathbb {F}_q[T]$ be the polynomial ring over $\mathbb {F}_q$ , a finite field with q elements. Let $K=\mathbb {F}_q(T)$ and $K_{\infty }=\mathbb {F}_q((1/T))$ denote the quotient field of A and the completion of K at $\infty =(1/T)$ , respectively. Let $\mathbb {C}_{\infty }$ be the completion of an algebraic closure of $K_{\infty }$ and let $A_+$ be the set of monic elements in A. We write $\Omega =\mathbb {C}_{\infty }\setminus K_{\infty }$ for the Drinfeld upper-half plane. Let $|\cdot |$ be the absolute value of $\mathbb {C}_{\infty }$ normalised by $|T|=q$ .

Let F be a field containing K and let E be a finite Galois extension of F. To find a completely normal element in $E/F$ , we use Artin’s argument in [Reference Artin1]. Let L be any intermediate field of $E/F$ and set $G=\text {Gal}(E/L) =\{\sigma _1=1, \ldots , \sigma _s\}$ . Let g be a primitive element of E over F and let $\alpha (x)$ be the minimal polynomial for g over L with $\deg \alpha (x)=s$ . For each $\sigma \in G$ , set $\beta _{\sigma }(x)=\alpha (x)/(x-g^{\sigma })$ . Moreover, let

$$ \begin{align*} D(x)=\det(\beta_{\sigma_i\sigma_j^{-1}}(x)). \end{align*} $$

From the definition of $\beta _{\sigma }$ ,

$$ \begin{align*} D(g)= \begin{vmatrix} \beta_1(g) & 0 & \cdots & 0\\ 0 & \beta_1(g) & \cdots & 0\\ \vdots & \vdots & \ddots & \vdots\\ 0 & 0 & \cdots & \beta_1(g) \end{vmatrix} =\beta_1(g)^s\ne 0, \end{align*} $$

which implies that $D(x)$ is not zero.

We have the following criterion for completely normal elements in $E/F$ .

Theorem 2.1. Let g be a primitive element of E over F. If $D(T^m)$ is nonzero for a positive integer m, then

$$ \begin{align*} \frac{1}{T^m-g} \end{align*} $$

is completely normal in $E/F$ .

Proof. First, $\det ((T^m-g^{\sigma _i\sigma _j^{-1}})^{-1})$ is nonzero because

$$ \begin{align*} D(T^m)=\alpha (T^m)^s\det\bigg(\frac{1}{T^m-g^{\sigma_i\sigma_j^{-1}}}\bigg). \end{align*} $$

By setting $J_m=(T^m-g)^{-1}$ , we prove that the $J_m^{\sigma }\ (\sigma \in G)$ are linearly independent over L. Let $\Sigma _{\sigma \in G}x_{\sigma }J_m^{\sigma }=0\ (x_{\sigma }\in L)$ . By letting $\tau ^{-1}\in G$ act in this equation,

$$ \begin{align*} \sum_{i=1}^sx_{\sigma_i}J_m^{\sigma_i\sigma_j^{-1}}=0\quad (j=1, \ldots , s) \end{align*} $$

for $\tau =\sigma _1, \ldots , \sigma _s$ . As the determinant of the coefficients of these equations is $\det ((T^m-g^{\sigma _i\sigma _j^{-1}})^{-1})$ , it follows that $x_{\sigma }=0$ for $\sigma \in G$ .

3. Overview of A-lattices and Drinfeld A-modules

We present an overview of A-lattices and Drinfeld A-modules. Further details can be found from Goss [Reference Goss7] and Rosen [Reference Rosen15]. A rank $r\ A$ -lattice $\Lambda $ in $\mathbb {C}_{\infty }$ is a finitely generated A-submodule of rank r in $\mathbb {C}_{\infty }$ that is discrete in the topology of $\mathbb {C}_{\infty }$ . For such an A-lattice $\Lambda $ , we define the product

$$ \begin{align*} e_{\Lambda}(z)=z\prod_{0\ne\lambda\in\Lambda} \bigg( 1-\frac{z}{\lambda}\bigg). \end{align*} $$

This product converges uniformly on bounded sets in $\mathbb {C}_{\infty }$ and defines a map $e_\Lambda $ such that $e_{\Lambda }: \mathbb {C}_{\infty }\to \mathbb {C}_{\infty }$ . The map $e_{\Lambda }$ has the following properties:

1. (E1) $e_{\Lambda }$ is entire in the rigid analytic sense and is surjective;

2. (E2) $e_{\Lambda }$ is $\mathbb {F}_q$ -linear and $\Lambda $ -periodic;

3. (E3) $e_{\Lambda }$ has simple zeros at the points of $\Lambda $ and no other zeros.

For every $a\in A$ , there exists a unique polynomial $\phi _a=\phi _a^{\Lambda }$ of the form $\sum l_i(a)z^{q^i}$ such that $\phi _a(e_{\Lambda }(z))=e_{\Lambda }(az)$ . Let $\tau =z^q$ and $\mathbb {C}_{\infty }\{\tau \}$ be the noncommutative ring in $\tau $ with the commutation rule $c^q\tau =\tau c$ ( $c\in \mathbb {C}_{\infty }$ ). There exists a unique positive integer r such that

$$ \begin{align*} \phi_a=\sum_{i=0}^{r\deg a}l_i(a)\tau^i\quad (l_0(a)=a, l_{r\deg a}(a)\ne 0) \end{align*} $$

for any $a\in A\setminus \{ 0\}$ . The map $\phi : A\to \mathbb {C}_{\infty }\{\tau \}$ , $a\mapsto \phi _a$ is then called a Drinfeld A-module of rank r over $\mathbb {C}_{\infty }$ . Because $\phi $ is an $\mathbb {F}_q$ -linear ring homomorphism, the values $\phi _a\ (a\in A)$ are determined by $\phi _T$ . The rank one Drinfeld A-module $\rho : A\to \mathbb {C}_{\infty }\{\tau \}$ defined by $\rho _T(z)=Tz+z^q$ is called the Carlitz module and the rank one A-lattice $L=\overline {\pi }A$ corresponding to $\rho $ is analogous to $2\pi i\mathbb {Z}$ . It is well known that there is a one-to-one correspondence between the set of A-lattices of rank r and the set of Drinfeld A-modules of rank r over $\mathbb {C}_{\infty }$ . This correspondence is given by $\phi _a(e_{\Lambda }(z)) = e_{\Lambda }(az)$ for all $a\in A$ .

4. Siegel functions

This section discusses Siegel functions.

4.1. Basic results

For $\omega \in \Omega $ , let $\Lambda _{\omega }=A\omega +A$ be the rank two A-lattice. For the rank two Drinfeld A-module $\phi ^{\Lambda _{\omega }} : A\to \mathbb {C}_{\infty }\{\tau \}$ corresponding to $\Lambda _{\omega }$ ,

(4.1) $$ \begin{align} \phi_T^{\Lambda_{\omega}}=T+g(\omega)\tau +\Delta (\omega)\tau^2. \end{align} $$

The function $\Delta $ is called the Drinfeld discriminant function and is a Drinfeld cusp form of weight $q^2-1$ for the Drinfeld modular group $\Gamma (1) =GL_2(A)$ . Let $\rho $ and $L=\overline {\pi }A$ be the Carlitz module and the corresponding rank one A-lattice, respectively. Considering $\rho _a\ (a\in A)$ to be a polynomial in x, we set

$$ \begin{align*} f_a(x)=\rho_a(x^{-1})x^{|a|}, \end{align*} $$

which is called the ath inverse cyclotomic polynomial. Let $t(\omega )=e_L(\overline {\pi }\omega )^{-1}$ . Gekeler [Reference Gekeler5] established a product formula for $\Delta $ .

Theorem 4.1 (Gekeler)

The function $\Delta $ has the product expansion

$$ \begin{align*} \Delta (\omega)= -\overline{\pi}^{q^2-1}t^{q-1} \prod_{0\ne a\in A}f_a(t)^{q^2-1} \end{align*} $$

with a positive radius of convergence for t.

Let

$$ \begin{align*} \eta (\omega)=\overline{\pi}t^{{1}/{(q+1)}} \prod_{0\ne a\in A}f_a(t). \end{align*} $$

Thus, $\eta ^{q^2-1}=-\Delta $ .

We take $n\in A_+$ with $\deg n>0$ . For $u=(u_1, u_2)\in (n^{-1}A/A)^2$ , let

$$ \begin{align*} e_u(\omega)=e_{\Lambda_{\omega}}(u_1\omega +u_2). \end{align*} $$

The Siegel function $g_u$ is formally defined as

$$ \begin{align*} g_u(\omega)=e_u(\omega)\eta (\omega). \end{align*} $$

The group $\Gamma (1)$ acts on $\Omega $ by $\sigma \omega =(a\omega +b)(c\omega +d)^{-1}$ for $\sigma =(\begin {smallmatrix}a & b\\ c & d\end {smallmatrix})\in \Gamma (1)$ and ${\omega \in \Omega} $ . The principal congruence subgroup of level n is

$$ \begin{align*} \Gamma (n)=\bigg\{\sigma\in\Gamma (1)\ \bigg|\ \sigma\equiv \begin{pmatrix}1 & 0\\ 0 & 1\end{pmatrix}\ (\text{mod}\ n)\bigg\}. \end{align*} $$

A congruence subgroup with conductor n of $\Gamma (1)$ is a subgroup $\Gamma $ containing $\Gamma (n)$ . For this congruence subgroup $\Gamma $ , the rigid analytic space $\Gamma \setminus \Omega $ is endowed with a smooth affine algebraic curve over $\mathbb {C}_{\infty }$ . The curve $X_{\Gamma }$ , which is a smooth projective model of $\Gamma \setminus \Omega $ , is the Drinfeld modular curve for $\Gamma $ . Let $X(n)$ and $X(1)$ be Drinfeld modular curves for $\Gamma (n)$ and $\Gamma (1)$ , respectively. In addition, let $\mathbb {C}_{\infty }(X(n))$ and $\mathbb {C}_{\infty }(X(1))$ be the meromorphic function fields of $\Gamma (n)$ and $\Gamma (1)$ , respectively. The group $\Gamma (1)$ acts on $\mathbb {C}_{\infty }(X(n))$ by $h^{\sigma }(\omega )=h(\sigma \omega )$ for $h\in \mathbb {C}_{\infty }(X(n))$ and $\sigma \in \Gamma (1)$ . It is known that $\mathbb {C}_{\infty }(X(n))/\mathbb {C}_{\infty }(X(1))$ is a Galois extension with

(4.2) $$ \begin{align} \text{Gal}(\mathbb{C}_{\infty}(X(n))/\mathbb{C}_{\infty}(X(1)))\cong \Gamma (1)/Z(\mathbb{F}_q)\Gamma (n), \end{align} $$

whose order is $|n|^3\prod _{P|n}(1-{1}/{|P|^2}) $ , where $Z(\mathbb {F}_q)$ denotes the $\mathbb {F}_q$ -valued scalar matrices, and the product $\prod _{P|n}$ is taken over all monic irreducibles P that divide n. (See [Reference Gekeler6] for further details.)

Let $t_n(\omega )=e_L(\overline {\pi }\omega /n)^{-1}$ , which is a parameter at $\infty $ for $X(n)$ .

Proposition 4.2 (Gekeler [Reference Gekeler4])

Let $n\in A_+$ with $\deg n>0$ and $u=(n^{-1}s_1, n^{-1}s_2)\in (n^{-1}A/A)^2$ with $\deg s_1, \deg s_2<\deg n$ . Then, the following statements hold.

1. (i) The order of $g_u(\omega )$ at $t_n$ is given by

   $$ \begin{align*} \mathrm{ord}_{t_n}g_u(\omega)=|n|\bigg(\frac{1}{q+1}-\frac{|s_1|}{|n|}\bigg). \end{align*} $$

2. (ii) For $\sigma \in \Gamma (1)$ , $g_u^{\sigma }=g_{u\sigma }$ .

3. (iii) For a subset S of $(n^{-1}A/A)^2$ , the product $\prod _{u\in S}g_u^{m(u)}$ belongs to $\mathbb {C}_{\infty }(X(n))$ if and only if $\sum _{u\in S}m(u)\equiv 0\ (\mathrm {mod}\ q+1)$ .

We establish the following product formula for the Siegel function $g_u$ .

Theorem 4.3 (A product formula)

Let $n\in A_+$ with $\deg n>0$ . For $u=(n^{-1}s_1, n^{-1}s_2) \in (n^{-1}A/A)^2$ , $g_u$ has the product expansion

$$ \begin{align*} g_u(\omega) & = -t_n^{{|n|}/{(q+1)}-|s_1|}f_n(t_n)^{-{1}/{(q+1)}} (f_{-s_1}(t_n)-e_L(\overline{\pi}s_2/n)t_n^{|s_1|})\\ & \quad\times \prod_{0\ne a\in A}(f_{na-s_1}(t_n)-e_L(\overline{\pi}s_2/n) t_n^{|na|}) \end{align*} $$

with a positive radius of convergence for $t_n$ .

Remark 4.4. The classical Siegel function $s_{\nu }(z)$ introduced in Section 1 has the following product expansion:

$$ \begin{align*} s_{\nu}(z)=-e^{\pi i\nu_2(\nu_1-1)}q^{B_2(\nu_1)/2}(1-q^{\nu_1}e^{2\pi i\nu_2}) \prod_{m=1}^{\infty}(1-q^{m+\nu_1}e^{2\pi i\nu_2})(1-q^{m-\nu_1}e^{-2\pi i\nu_2}), \end{align*} $$

where $q=e^{2\pi iz}$ and $B_2(x)=x^2-x+1/6$ .

4.2. Proof of Theorem 4.3

Using [Reference Gekeler4, (2.1)],

$$ \begin{align*} e_u(\omega)= \overline{\pi}^{-1}t_n^{-|s_1|} (f_{s_1}(t_n)+e_L(\overline{\pi}s_2/n)t_n^{|s_1|}) \prod_{0\ne a\in A} \frac{f_{na-s_1}(t_n)-e_L(\overline{\pi}s_2/n)t_n^{|na|}}{f_{na}(t_n)}. \end{align*} $$

As $t=t_n^{|n|}/f_n(t_n)$ , for $a\in A\setminus \{ 0\}$ ,

$$ \begin{align*} f_a(t) = \rho_{na}(e_L(\overline{\pi}\omega /n))t^{|a|}= \rho_{na}(e_L(\overline{\pi}\omega /n))\bigg(\frac{t_n^{|n|}}{f_n(t_n)}\bigg)^{|a|} = \frac{f_{na}(t_n)}{f_n(t_n)^{|a|}}, \end{align*} $$

which yields

$$ \begin{align*} \eta (\omega) =\overline{\pi}t_n^{{|n|}/{(q+1)}}f_n(t_n)^{-{1}/{(q+1)}} \prod_{0\ne a\in A}\frac{f_{na}(t_n)}{f_n(t_n)^{|a|}}. \end{align*} $$

Therefore,

$$ \begin{align*} g_u(\omega) & = -t_n^{{|n|}/{(q+1)}-|s_1|}f_n(t_n)^{-{1}/{(q+1)}} (f_{-s_1}(t_n)-e_L(\overline{\pi}s_2/n)t_n^{|s_1|})\\ & \quad\times \prod_{0\ne a\in A} \frac{f_{na-s_1}(t_n)-e_L(\overline{\pi}s_2/n)t_n^{|na|}}{f_n(t_n)^{|a|}}. \end{align*} $$

To simplify this expression, the following lemma is required.

Lemma 4.5. We have

$$ \begin{align*} \prod_{a\in A}f_n(t_n)^{|a|}=1. \end{align*} $$

Proof. For $a\in A$ , let

$$ \begin{align*} W_a(t_n)=f_n(t_n)^{-|a|}\prod_{c\in\mathbb{F}_q}f_n(t_n)^{|aT+c|}. \end{align*} $$

Because

$$ \begin{align*} f_n(t_n)^{|a|}W_a(t_n)=\prod_{c\in\mathbb{F}_q}f_n(t_n)^{|aT+c|} \quad\mbox{and}\quad \prod_{\deg a\leq n}f_n(t_n)^{|a|}W_a(t_n) =\prod_{\deg a\leq n+1}f_n(t_n)^{|a|}, \end{align*} $$

it follows that

$$ \begin{align*} \prod_{0\ne a\in A}f_n(t_n)^{|a|}W_a(t_n) =\prod_{0\ne a\in A}f_n(t_n)^{|a|}, \end{align*} $$

which yields $\prod _{0\ne a\in A}W_a(t_n)=1$ . From this,

$$ \begin{align*} \prod_{0\ne a\in A}f_n(t_n)^{|a|}= \prod_{0\ne a\in A}\prod_{c\in\mathbb{F}_q}f_n(t_n)^{|aT+c|}= \bigg(\prod_{0\ne a\in A}f_n(t_n)^{|a|}\bigg)^{q^2}, \end{align*} $$

which yields $(\prod _{0\ne a\in A}f_n(t_n)^{|a|})^{q^2-1}=1$ . Noting that $\prod _{0\ne a\in A}f_n(0)^{|a|}=1$ , we have $\prod _{0\ne a\in A}f_n(t_n)^{|a|}=1$ .

From this lemma, the proof of Theorem 4.3 is completed.

5. Normal bases for Drinfeld modular function fields

In this section, we construct the completely normal elements in Drinfeld modular function fields.

5.1. The primitive element $h_n$

Using Siegel functions, we construct a primitive element of $\mathbb {C}_{\infty }(X(n))$ over $\mathbb {C}_{\infty }(X(1))$ .

Definition 5.1. We set

$$ \begin{align*} h_n=\frac{1}{g_{(0,n^{-1})}^{2q+1}g_{(n^{-1},0)}}. \end{align*} $$

The function $h_n$ has the following properties.

Proposition 5.2.

1. (i) $h_n\in \mathbb {C}_{\infty }(X(n))$ .

2. (ii) $h_n(\lambda \omega )=\lambda h_n(\omega )$ for $\lambda \in \mathbb {F}_q^{\ast }$ .

Proof.

1. (i) This follows from Proposition 4.2(iii).

2. (ii) For $\lambda \in \mathbb {F}_q^{\ast }$ and $a\in A\setminus \{ 0\}$ ,

   $$ \begin{align*} t(\lambda\omega)=\lambda^{-1}t(\omega),\quad f_a(t(\lambda\omega))=f_a(t(\omega)), \quad \eta (\lambda\omega)=\lambda^{-{1}/{(q+1)}}\eta (\omega),\\ g_{(0,n^{-1})}(\lambda\omega)=\lambda^{-{1}/{(q+1)}}g_{(0,n^{-1})}(\omega),\quad g_{(n^{-1},0)}(\lambda\omega)=\lambda^{{q}/{(q+1)}}g_{(n^{-1},0)}(\omega). \end{align*} $$
   Thus, we obtain property (ii).

Lemma 5.3. For $\sigma \in \Gamma (1)$ , $\mathrm {ord}_{t_n}(h_n^{\sigma }/h_n)\geq 0$ . Equality holds if and only if

$$ \begin{align*} \sigma\equiv\begin{pmatrix}a_1 & \ast \\ 0 & d_1\end{pmatrix}\ (\mathrm{mod}\ n), \end{align*} $$

where $a_1, d_1\in \mathbb {F}_q^{\ast }$ .

Proof. Let $\sigma \equiv (\begin {smallmatrix}a_1 & b_1\\ c_1 & d_1\end {smallmatrix})\ (\text {mod}\ n)$ , where $a_1, b_1, c_1, d_1\in A,\ \deg a_1, \deg b_1, \deg c_1, \deg d_1 < \deg n$ . Using Proposition 4.2,

$$ \begin{align*} \text{ord}_{t_n}(h_n^{\sigma}/h_n) =(2q+1)|c_1|+(|a_1|-1). \end{align*} $$

Since $a_1d_1-b_1c_1\in \mathbb {F}_q^{\ast }$ , either $a_1\ne 0$ or $c_1\ne 0$ .

1. (i) When $a_1\ne 0$ and $c_1\ne 0$ , $(2q+1)|c_1|+(|a_1|-1)>0$ .

2. (ii) When $a_1=0$ and $c_1\ne 0$ , $(2q+1)|c_1|+(|a_1|-1)\geq 2q>0$ .

3. (iii) When $a_1\ne 0$ and $c_1=0$ , $(2q+1)|c_1|+(|a_1|-1)=|a_1|-1\geq 0$ ,

which yields the first part of the lemma.

For the latter part of the lemma, we use item (iii). Equality holds if and only if $c_1=0, a_1\in \mathbb {F}_q^{\ast }$ , which is equivalent to $c\in nA$ and $a_1, d_1\in \mathbb {F}_q^{\ast }$ .

Proposition 5.4. The function $h_n$ generates $\mathbb {C}_{\infty }(X(n))$ over $\mathbb {C}_{\infty }(X(1))$ .

Proof. We assume that $\sigma =(\begin {smallmatrix}a & b\\ c & d\end {smallmatrix})\in \Gamma (1)$ leaves $h_n$ fixed. Because $\text {ord}_{t_n}h_n^{\sigma }=\text {ord}_{t_n}h_n$ , Lemma 5.3 implies that

$$ \begin{align*} a\equiv a_1,\ c\equiv 0,\ d\equiv d_1\ (\text{mod}\ n),\quad a_1, d_1\in\mathbb{F}_q^{\ast}. \end{align*} $$

For $\tau =(\begin {smallmatrix}0 & -1\\ 1 & 0\end {smallmatrix})$ , $\text {ord}_{t_n}h_n^{\sigma \tau } =\text {ord}_{t_n}h_n^{\tau }$ , which yields

$$ \begin{align*} |n|\bigg[(2q+1)\bigg(\frac{1}{|n|}-\frac{1}{q+1}\bigg) +\bigg(\frac{|b_1|}{|n|}-\frac{1}{q+1}\bigg)\bigg] =|n|\bigg[(2q+1)\bigg(\frac{1}{|n|}-\frac{1}{q+1}\bigg)+\bigg(-\frac{1}{q+1}\bigg)\bigg] \end{align*} $$

by Proposition 4.2. In this expression, $b_1\in A$ is defined by $b\equiv b_1\ (\text {mod}\ n), \deg b_1<\deg n$ . Hence, $b_1=0$ and $\sigma \equiv (\begin {smallmatrix}a_1 & 0\\ 0 & d_1\end {smallmatrix}\!) \ (\text {mod}\ n)$ . By letting $\gamma =\sigma (\begin {smallmatrix}a_1^{-1} & 0\\ 0 & d_1^{-1}\end {smallmatrix}\!)$ , we observe that $\gamma \in \Gamma (n)$ . Using Proposition 4.2,

$$ \begin{align*} h_n=h_n^{\sigma}=h_n^{\gamma\big(\begin{smallmatrix}a_1 & 0\\ 0 & d_1\end{smallmatrix}\!\big)} =h_n^{\big(\begin{smallmatrix}a_1 & 0\\ 0 & d_1\end{smallmatrix}\!\big)} =\frac{a_1}{d_1}h_n. \end{align*} $$

Hence, $\sigma = \big (\begin {smallmatrix}a_1 & 0\\ 0 & a_1\end {smallmatrix}\!\big)\gamma \in Z(\mathbb {F}_q)\Gamma (n)$ . Using (4.2) and Galois theory, the proof of this proposition is completed.

Remark 5.5. It is known that

$$ \begin{align*} \mathbb{C}_{\infty}(X(1))=\mathbb{C}_{\infty}(j),\quad \mathbb{C}_{\infty}(X(n))=\mathbb{C}_{\infty}(j, f_u\ |\ u\in (n^{-1}A/A)^2), \end{align*} $$

where j is the modular function defined by $j(\omega )=g(\omega )^{q+1}/\Delta (\omega )$ using $g, \Delta $ in (4.1), and $f_u$ is the Fricke function defined by $f_u(\omega )=g(\omega )e_u(\omega )^{q-1}$ . From Proposition 5.4, we obtain $\mathbb {C}_{\infty }(X(n))=\mathbb {C}_{\infty }(j, h_n)$ .

The following lemma is required in the next subsection.

Lemma 5.6. Let l be a positive integer. If

(5.1) $$ \begin{align} q^{-2l}<|t_n|\leq q^{-l-{1}/{(q-1)}}, \end{align} $$

then

$$ \begin{align*} |h_n^{-1}(\sigma\omega)|< q^{4ql|n|+6} \end{align*} $$

for any $\sigma \in \Gamma (1)$ .

Proof. Let $\sigma \equiv (\begin {smallmatrix}a_1 & b_1\\ c_1 & d_1\end {smallmatrix})\ (\text {mod}\ n)$ , where $a_1, b_1, c_1, d_1\in A,\ \deg a_1, \deg b_1, \deg c_1, \deg d_1 <\deg n$ . By Proposition 4.2(ii) and Theorem 4.3,

$$ \begin{align*} h_n^{-1}(\sigma\omega) & = t_n^{(|n|-|a_1|)+(|n|-|c_1|)-2q|c_1|}f_n(t_n)^{-2}\\& \quad \times (f_{-c_1}(t_n)-e_L(\overline{\pi}d_1/n)t_n^{|c_1|})^{2q+1} (f_{-a_1}(t_n)-e_L(\overline{\pi}b_1/n)t_n^{|a_1|})\\& \quad \times \prod_{0\ne a\in A} (f_{na-c_1}(t_n)-e_L(\overline{\pi}d_1/n)t_n^{|na|})^{2q+1} (f_{na-a_1}(t_n)-e_L(\overline{\pi}b_1/n)t_n^{|na|}). \end{align*} $$

Gekeler [Reference Gekeler5, Lemma 1] proved that if $t_n$ satisfies (5.1) and $a\in A\setminus \{ 0\}$ with $\deg a=d$ , then $|f_a(t_n)-1|\leq (q^{-l})^{q^{d-1}(q-1)}$ . For $e_L(\overline {\pi }d_1/n)$ ,

$$ \begin{align*} |e_L(\overline{\pi}d_1/n)|=|\overline{\pi}e_A(d_1/n)| =q^{{q}/{(q-1)}+\deg d_1-\deg n}\leq q^{{1}/{(q-1)}}. \end{align*} $$

Similarly, we find that $|e_L(\overline {\pi }b_1/n)|\leq q^{{1}/{(q-1)}}$ . Therefore,

$$ \begin{align*} |h_n^{-1}(\sigma\omega)|\leq |t_n|^{-2q|c_1|} |f_{-c_1}(t_n)-e_L(\overline{\pi}d_1/n)t_n^{|c_1|}|^{2q+1} |f_{-a_1}(t_n)-e_L(\overline{\pi}b_1/n)t_n^{|a_1|}|. \end{align*} $$

We have

$$ \begin{align*} |t_n|^{-2q|c_1|} & < (q^{-2l})^{-2q|n|},\\ |f_{-c_1}(t_n)-e_L(\overline{\pi}d_1/n)t_n^{|c_1|}| ^{2q+1}& \leq q^{{(2q+1)}/{(q-1)}},\\ |f_{-a_1}(t_n)-e_L(\overline{\pi}b_1/n)t_n^{|a_1|}| & \leq q^{{1}/{(q-1)}}, \end{align*} $$

which yields the lemma.

5.2. Completely normal elements

Using $h_n$ , we construct completely normal elements in $\mathbb {C}_{\infty }(X(n))/\mathbb {C}_{\infty }(X(1))$ . Let $r=[\mathbb {C}_{\infty }(X(n)):\mathbb {C}_{\infty }(X(1))]$ .

Theorem 5.7. Let g be a primitive element of $\mathbb {C}_{\infty }(X(n))$ over $\mathbb {C}_{\infty }(X(1))$ and let $0<\delta _2<\delta _1<1$ . We assume that there exists a positive constant $M\geq 1$ such that ${|g(\omega )|<M}$ for $\delta _2<|t_n(\omega )|<\delta _1$ . If $m>\log _qM^{r(r-1)}$ , then

$$ \begin{align*} \frac{1}{T^m-g} \end{align*} $$

is completely normal in $\mathbb {C}_{\infty }(X(n))/\mathbb {C}_{\infty }(X(1))$ .

Proof. Let L be any intermediate field of $\mathbb {C}_{\infty }(X(n))/\mathbb {C}_{\infty }(X(1))$ , and set

$$ \begin{align*}G=\text{Gal}(\mathbb{C}_{\infty}(X(n))/L) =\{\sigma_1=1, \ldots , \sigma_s\}.\end{align*} $$

If $\alpha (x)$ is the minimal polynomial for g over L, then $\deg \alpha (x)=s$ . For each ${\sigma \in G}$ , set $\beta _{\sigma }(x)=\alpha (x)/(x-g^{\sigma })$ . Moreover, let $D(x)=\det (\beta _{\sigma _i\sigma _j^{-1}}(x))$ . For $\omega \in \Omega $ with ${\delta _2<|t_n(\omega )|<\delta _1}$ , we set

$$ \begin{align*} \beta_{\sigma}(x,\omega) = \prod_{\gamma\in G\setminus\{\sigma\}} (x-g^{\gamma}(\omega)) \quad\mbox{and}\quad D(x,\omega) = \text{det}(\beta_{\sigma_i\sigma_j^{-1}}(x,\omega)). \end{align*} $$

As the coefficients of $\beta _{\sigma }(x,\omega )$ are sums of products of $g^{\gamma }(\omega )\ (\gamma \in G\setminus \{\sigma \})$ , the absolute value of each of these coefficients is not greater than $M^{s-1}$ . Hence, the absolute value of each coefficient of $D(x,\omega )$ is not greater than $M^{s(s-1)}$ . If $m>\log _qM^{r(r-1)}$ , then $q^m>M^{s(s-1)}$ . When $D(x, \omega )=\sum _{i=0}^{s(s-1)}a_ix^i$ , the absolute value of a nonzero $a_i$ satisfies $|a_i|\leq M^{s(s-1)}<q^m$ , which implies that $q^{mi}\leq |a_iT^{mi}|<q^{m(i+1)}$ . Thus, $|D(T^m,\omega )|$ is nonzero, implying that $D(T^m)$ is also nonzero. Therefore, the theorem follows from Theorem 2.1.

Definition 5.8. For a positive integer m, we set

$$ \begin{align*} H_{n,m}=\frac{h_n}{T^mh_n-1}. \end{align*} $$

The function $H_{n,m}$ belongs to $\mathbb {C}_{\infty }(X(n))$ . The following theorem is a consequence of Theorem 5.7.

Theorem 5.9. If $m>2r(r-1)(2q|n|+1)$ , then the element $H_{n,m}$ is completely normal in $\mathbb {C}_{\infty }(X(n))/\mathbb {C}_{\infty }(X(1))$ .

Proof. Using Proposition 5.4, $h_n^{-1}$ is a primitive element of $\mathbb {C}_{\infty }(X(n))$ over $\mathbb {C}_{\infty }(X(1))$ . Taking a positive integer $l>2$ , let $\delta _1=q^{-l-2}$ and $\delta _2=q^{-2l}$ . If $\delta _2<|t_n|<\delta _1$ , then we have $|h_n^{-1}(\sigma \omega )|<q^{4ql|n|+6}$ for $\sigma \in \Gamma (1)$ using Lemma 5.6. Applying Theorem 5.7 for $M=q^{4ql|n|+6}$ and $g=h_n^{-1}$ , we obtain the theorem.

6. Normal bases for cyclotomic function fields

In this section, we construct completely normal elements in cyclotomic function fields and their maximal real subfields.

Let $\rho $ be the Carlitz module. For $n\in A_+$ , let $\rho [n]=\{\alpha \in \mathbb {C}_{\infty }\ |\ \rho _n(\alpha )=0\}$ be the set of Carlitz n-torsion points. The set $\rho [n]$ is a cyclic A-module and its generator is called the primitive Carlitz n-torsion point. The minimal polynomial $\Phi _n(x)$ for any primitive n-torsion point over K is called the Carlitz nth cyclotomic polynomial. The polynomials $\rho _n(x)$ and $\Phi _n(x)$ have degrees $q^{\deg n}$ and $\varphi (n)$ , respectively, where $\varphi (n):=\#(A/nA)^{\ast }$ . (For further details on these polynomials, we refer to [Reference Blessenohl and Johnson3].) For the primitive Carlitz n-torsion point $\lambda _n$ , let $K_n=K(\lambda _n)$ be the field generated over K by adjoining $\lambda _n$ . If $\sigma \in \text {Gal}(K_n/K)$ , then $\sigma (\lambda _n)$ is another primitive Carlitz n-torsion point. Hence, there exists $a\in A$ with $\gcd (a, n)=1$ such that $\sigma (\lambda _n)=\rho _a(\lambda _n)$ . Let $\epsilon _a=\sigma $ . The correspondence $\epsilon _a\mapsto a$ induces the isomorphism $\text {Gal}(K_n/K)\widetilde {\to } (A/nA)^{\ast }$ (see [Reference Rosen15, Theorem 12.8]).

Theorem 6.1. Let F be a finite Galois extension of K of degree r contained in $\mathbb {C}_{\infty }$ and let $\mu $ be a primitive element of F over K with $|\mu |\leq M$ , where $M\geq 1$ is a constant. If $m>\log _qM^{r(r-1)}$ , then

$$ \begin{align*} \frac{1}{T^m-\mu} \end{align*} $$

is completely normal in $F/K$ .

For a completely normal element in $K_n/K$ , we have the following result.

Theorem 6.2. For a monic element $n\in A_+$ with $\deg n>0$ , let $K_n$ be the cyclotomic function field determined by n. For any positive integer m,

$$ \begin{align*} \frac{1}{T^m-e_L(\overline{\pi}/n)} \end{align*} $$

is completely normal in $K_n/K$ .

Let $K_n^+$ be the fixed field of $\{\epsilon _a\in \text {Gal}(K_n/K)\ |\ a\in \mathbb {F}_q^{\ast }\}$ . This field is referred to as the maximal real subfield of $K_n$ . For a completely normal element in $K_n^+/K$ , we have the following result.

Theorem 6.3. Notation being as in Theorem 6.2, we let $r=[K_n^+:K]$ . If $m>r(r-1)$ , then

$$ \begin{align*} \frac{1}{T^m-e_L(\overline{\pi}/n)^{q-1}} \end{align*} $$

is completely normal in $K_n^+/K$ .

Remark 6.4. For $n\kern1.3pt{\in}\kern1.3pt A_+$ with $\deg n\kern1.3pt{>}\kern1.3pt0$ , let $R_n\kern1.3pt{=}\kern1.3pt\{a\kern1.3pt{\in}\kern1.3pt A_+\ |\ \deg a\kern1.3pt{<}\kern1.3pt\deg n, \text {gcd}(a, n)\kern1.3pt{=}\kern1.3pt1\}$ . In [Reference Hamahata9], we proved that

$$ \begin{align*} G_k(e_L(\overline{\pi}a/n)^{-1})\quad (a\in R_n) \end{align*} $$

are linearly independent over K for any positive integer k, where $G_k(x)$ is the kth Goss polynomial of $L=\overline {\pi }A$ . It is known that $G_{q-1}(x)=x^{q-1}$ . As $\{1/e_L(\overline {\pi }a/n)^{q-1}\ |\ a\in R_n\}$ is contained in $K_n^+$ , $\{\epsilon _(1/e_L(\overline {\pi }/n)^{q-1})\ |\ a\in R_n\} =\{1/e_L(\overline {\pi }a/n)^{q-1}\ |\ a\in R_n\}$ is a normal basis for $K_n^+/K$ . This is analogous to the result provided by Okada [Reference Okada14].