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?