Condition for existence of mle with a cauchy distribution

Question

I'm solving some problem from statistical theory about maximum likelihood estimater.

Suppose $X_1,...,X_n$ are independent samples from $f(x;\alpha,\beta)$. ($n$ is an odd number.)

$$ f(x;\alpha,\beta) = \frac{1}{\pi\beta\Big[1+\big(\frac{x-\alpha}{\beta}\big)^2\Big]}, \alpha \in \mathbb{R}, \beta \in (0,\infty). $$

When does maximum likelihood estimater of $\alpha, \beta$ exist?