The same question was asked in Math StackExchange about 3 months ago. Since nobody has answered to it, I would like to post it here.

References:

Weil's Basic Number Theory(denoted by **BNT**).

Bourbaki's Commutative Algebra(denoted by **BCA**).

Let $K$ be a topological ring with an identity. Suppose every non-zero element of $K$ is invertible. Let $K^* = K - \{0\}$ be the multipilcative group of $K$ If the map $x \rightarrow x^{-1}$ is continuous on $K^*$, we say $K$ is a topological division ring. Suppose the topological space $K$ is non-discrete, Hausdorff and locally compact. Then we say, by abuse of terminology, $K$ is a locally compact division ring.

Let $K$ be a locally compact division ring. Then the aditive group $K$ is a locally compact group. Hence there exists a Haar measure $\mu$ on $K$. Let $a$ be an element of $K^*$. Then the map $x \rightarrow ax$ is an automorphism of the locally compact group $K$. Hence the map $X \rightarrow \mu(aX)$ defines an invariant measure on $K$ where $X$ is any measurable subset of $K$. Therefore there exists a constant $c \gt 0$ such that $\mu(aX) = c\mu(X)$ for every measurable subset $X$ such that $0 \lt \mu(X) \lt \infty$. We denote $c$ by $mod(a)$. We define $mod(0) = 0$.

$mod(a)$ can also be defined by the map $x \rightarrow xa$(see **BNT** or **BCA**).

Clearly $mod(ab) = mod(a)mod(b)$ for all $a, b \in K$.
The function $mod$ is continuous(see **BNT** or **BCA**).
The subset $\{x \in K|\ mod(x) \le d\}$ is compact for every real number $d \gt 0$(see **BNT**).

Locally compact division rings are classified as follows(see **BNT** or **BCA**).

The field of real numbers $\mathbb{R}$.

The field of complex numbers $\mathbb{C}$.

The field of Hamilton's quaternions $\mathbb{H}$.

Finite division algebras over the field of $p$-adic numbers.

Finite division algebras over the field of formal Laurent series over a finite field.

Here is my question. Is the following proposition true?

**Proposition**
Let $K$ be a locally compact division ring.
Let $\phi$ be a real valued function defined on $K$.
Suppose $\phi$ satisies the following conditons.

$\phi$ is continuous.

$\phi(x) \gt 0$ for all $x \neq 0$ and $\phi(0) = 0$.

$\phi(xy) = \phi(x)\phi(y)$ for all $x, y$.

Then there exists a real number $c \gt 0$ such that $\phi(x) = mod(x)^c$ for all $x$.

**Remark**
Let $K$ be a locally compact division ring.
Let $x \in K^*$.

If $K = \mathbb{R}$, then $mod(x) = |x|$.

If $K = \mathbb{C}$, then $mod(x) = |x|^2$.

If $K = \mathbb{H}$, then $mod(x) = |x|^4$.

If $K = \mathbb{Q}_p$, then $mod(x) = |x|_p$ where $|x|_p$ is the canonical absolute value, i.e. $|p|_p = 1/p$.