The following statement is taken from Cohn's 1995 book Skew Fields: Theory of General Division Rings (Cambridge EOM 57), proposition 2.3.4 on page 69, which is attributed to Köthe:

Let $k$ be a commutative field with an automorphism $\alpha$ and put $E = k(\!(t;\alpha)\!)$ [the division ring of $\alpha$-twisted Laurent series over $k$]. Given any automorphism $\beta$ of $k$ such that $\alpha\beta = \beta\alpha$, extend $\beta$ to $E$ by the rule $t^\beta = t$. Then $\beta$ is an inner automorphism of $E$ if and only if $\beta = \alpha^r$ for some $r\in\mathbb{Z}$.

Cohn adds:

For example, if $k = F(s)$, where $F$ is any field of characteristic $0$, and $\alpha\colon s\mapsto s+1$ the $\beta\colon s\mapsto s+\frac{1}{2}$ is an outer automorphism of $E = k(\!(t;\alpha)\!)$.

For completeness of MathOverflow, let me recall the definition of the division ring $E = k(\!(t;\alpha)\!)$ of $\alpha$-twisted (or "skew") Laurent series: its elements are of the form $\sum_{i=i_0}^{+\infty} t^i a_i$ with $a_i \in k$ [here I'm following Cohn's convention of using right-multiplication by $k$], addition being defined componentwise and multiplication by distributing and using the rule $a t^n = t^n a^{\alpha^n}$ [where, of course, $a^{\alpha^n}$ denotes the image of $a$ under the $n$-th power of $\alpha$].

The following statement is taken from Cohn's 1995 book Skew Fields: Theory of General Division Rings (Cambridge EOM 57), proposition 2.3.4 on page 69, which is attributed to Köthe:

Let $k$ be a commutative field with an automorphism $\alpha$ and put $E = k(\!(t;\alpha)\!)$ [the division ring of $\alpha$-twisted Laurent series over $k$]. Given any automorphism $\beta$ of $k$ such that $\alpha\beta = \beta\alpha$, extend $\beta$ to $E$ by the rule $t^\beta = t$. Then $\beta$ is an inner automorphism of $E$ if and only if $\beta = \alpha^r$ for some $r\in\mathbb{Z}$.

Cohn adds:

For example, if $k = F(s)$, where $F$ is any field of characteristic $0$, and $\alpha\colon s\mapsto s+1$ the $\beta\colon s\mapsto s+\frac{1}{2}$ is an outer automorphism of $E = k(\!(t;\alpha)\!)$.

For completeness of MathOverflow, let me recall the definition of the division ring $E = k(\!(t;\alpha)\!)$ of $\alpha$-twisted (or "skew") Laurent series: its elements are of the form $\sum_{i=i_0}^{+\infty} t^i c_i$ with $c_i \in k$ [here I'm following Cohn's convention of using right-multiplication by $k$], addition being defined componentwise and multiplication by distributing and using the rule $c t^n = t^n c^{\alpha^n}$ [where, of course, $c^{\alpha^n}$ denotes the image of $c$ under the $n$-th power of $\alpha$].

The following statement is taken from Cohn's 1995 book Skew Fields: Theory of General Division Rings (Cambridge EOM 57), proposition 2.3.4 on page 69, which is attributed to Köthe:

Let $k$ be a commutative field with an automorphism $\alpha$ and put $E = k(\!(t;\alpha)\!)$ [the division ring of $\alpha$-twisted Laurent series over $k$]. Given any automorphism $\beta$ of $k$ such that $\alpha\beta = \beta\alpha$, extend $\beta$ to $E$ by the rule $t^\beta = t$. Then $\beta$ is an inner automorphism of $E$ if and only if $\beta = \alpha^r$ for some $r\in\mathbb{Z}$.

Cohn adds:

For example, if $k = F(s)$, where $F$ is any field of characteristic $0$, and $\alpha\colon s\mapsto s+1$ the $\beta\colon s\mapsto s+\frac{1}{2}$ is an outer automorphism of $E = k(\!(t;\alpha)\!)$.

The following statement is taken from Cohn's 1995 book Skew Fields: Theory of General Division Rings (Cambridge EOM 57), proposition 2.3.4 on page 69, which is attributed to Köthe:

Let $k$ be a commutative field with an automorphism $\alpha$ and put $E = k(\!(t;\alpha)\!)$ [the division ring of $\alpha$-twisted Laurent series over $k$]. Given any automorphism $\beta$ of $k$ such that $\alpha\beta = \beta\alpha$, extend $\beta$ to $E$ by the rule $t^\beta = t$. Then $\beta$ is an inner automorphism of $E$ if and only if $\beta = \alpha^r$ for some $r\in\mathbb{Z}$.

Cohn adds:

For example, if $k = F(s)$, where $F$ is any field of characteristic $0$, and $\alpha\colon s\mapsto s+1$ the $\beta\colon s\mapsto s+\frac{1}{2}$ is an outer automorphism of $E = k(\!(t;\alpha)\!)$.

For completeness of MathOverflow, let me recall the definition of the division ring $E = k(\!(t;\alpha)\!)$ of $\alpha$-twisted (or "skew") Laurent series: its elements are of the form $\sum_{i=i_0}^{+\infty} t^i a_i$ with $a_i \in k$ [here I'm following Cohn's convention of using right-multiplication by $k$], addition being defined componentwise and multiplication by distributing and using the rule $a t^n = t^n a^{\alpha^n}$ [where, of course, $a^{\alpha^n}$ denotes the image of $a$ under the $n$-th power of $\alpha$].