F-singularities via alterations

We give characterizations of test ideals and $F$-rational singularities via (regular) alterations. Formally, the descriptions are analogous to standard characterizations of multiplier ideals and rational singularities in characteristic zero via log resolutions. Lastly, we establish Nadel-type vanishing theorems (up to finite maps) for test ideals, and further demonstrate how these vanishing theorems may be used to extend sections.