OLS Coefficient Confidence Intervals

Consider the simple linear regression model $Y_i = \beta_0 + \beta_1 x_i + \varepsilon_i$ for $i = 1, \ldots, n$, where the errors $\varepsilon_i$ are i.i.d. $N(0, \sigma^2)$. 1. Derive the distribution of the OLS estimator $\hat{\beta}_1$ and use it to construct a confidence interval for $\beta_1$. 2. What assumptions are required for the confidence interval to be valid? What happens if they are violated?