Here is the full problem:
Let $X_1,...,X_n$ be a random sample from a $N(\mu,\sigma^2)$ distribution. Let $\tau = \sigma^{-2}$, so we can write the distribution as $N(\mu,\tau^{-1})$.
Suppose the prior for $(\mu,\tau)$ has density $f(\mu,\tau) \propto 1$, $-\infty < \mu < \infty$, $0 < \tau < \infty$. Note that this is an improper prior density.
Show that the posterior density of $(\mu,\tau)$ is equal to:
$f(\mu, \tau|x_1,...,x_n) = f(\mu|\tau,x_1,...,x_n) f(\tau|x_1,...,x_n)\;.$
where
$f(\mu|\tau,x_1,...,x_n) \sim N(\bar{x},(\tau \ n)^{-1})\;$
and
$f(\tau|x_1,...,x_n) \sim \text{Gamma}(\frac{n-1}{2},(\frac{\sum_{i=1}^n(x_i-\bar{x})^2}{2})^{-1})$
I have tried to break down the problem using the standard Bayesian method, but I am having trouble figuring out how we can "split" the density up. Thanks in advance.