Let $F$ be a local field of characteristic 0, and $f:F\rightarrow \mathbb{C}$ be an integrable function. Is the following formulation valid?


$
\int\_{F^\times}f(x^2) d^\times x=\int\_{F^{\times 2}}f(x) d^\times x
$


where $d^\times x$ is a chosen multiplicative measure on $F^\times$.


It seems if $F=\mathbb{R}$, this is true. But is it true for any local field?



A question on Haar measure on local field.
Since the subgroup of squares is open, the restriction of Haar measure from the full multiplicative group to the squares is a Haar measure on the subgroup of squares, and Haar measure is unique up to scalars.


