Why Divide by n − 1?

Let $X_1, \dots, X_n$ be i.i.d. with mean $\mu$ and variance $\sigma^2$, and let $\bar X = \frac1n\sum_i X_i$.

Show that

$$S^2 = \frac{1}{n-1}\sum_{i=1}^n (X_i - \bar X)^2$$

is an unbiased estimator of $\sigma^2$, i.e. $\mathbb{E}[S^2] = \sigma^2$.