ALGEBRAIC GROUPS Disclaimer: There are millions of errors in these notes! 1. Some algebraic geometry The subject of algebraic groups depends on the interaction between alge-

S = k[T1, T2, . . . , Tn](= k[T ] for short) as k-valued functions on the space kn: if v = (v1, . . . , vn) ∈ kn and f = f(T1, . . . , Tn) then f(v) = f(v1, . . . , vn). We call v ∈ kn a zero of f if f(v) = 0. Given an ideal I ⊂ S, let V (I) denote the set of all common zeros of all functions in I, i.e. V (I) = {v ∈ k | f(v) = 0 for all f ∈ I}. Given a subset X ⊂ kn, let I(X) denote the ideal of all functions f ∈ S vanishing on all of X, i.e.