# Non-existence of unbiased estimator of $\sigma^2$ based on $X\sim N(\mu,\sigma^2)$ when $\mu$ is unknown

I attempted this via a proof of contradiction:

Assume via contradiction that there exists a $$\delta(X)$$ such that $$E[\delta(X)]=\sigma^2$$ $$\forall \mu \in \Theta$$.

I know that for a fixed $$\sigma$$ that $$X$$ is a complete sufficient statistic for $$\mu$$. My goal is to find a function of $$X$$ whose expected value is $$0$$ for a fixed $$\sigma^2$$ and which is not equal to $$0$$ a.e.

Fix $$\sigma_0$$ then $$X$$ is a complete sufficient static of $$\mu$$ . By assumption there exists a $$\delta(X)$$ which is an unbiased estimator for $$\sigma_0^2$$, $$\forall \mu$$.

By completeness there exists a unique $$\delta^{*}(X)$$ such that $$E[\delta^{*}(X)]=g(\mu)$$, $$\forall \sigma^2$$, where $$\delta^{*}(X)$$ cannot be a constant. Then consider $$\delta^{'}(X)=\delta(X)-\delta^{*}(X)$$ whose $$E[\delta^{'}(X)=\delta(X)-\delta^{*}(X)]=g(u)-\sigma_0^2$$. If we set $$g(u)=\sigma_0^2$$ then we have a contradiction since $$\delta^{'}$$ is non zero and is an estimator of 0.

I know my argument is not correct, but I am just lost as to how to prove this. Any help?