Abstract
We begin with the basic facts concerning the irreducible components of algebraic sets in general, and the irreducible component of the neutral element in an algebraic group. The main result of Section 2 is the fact that algebraic subgroups are determined by their semi-invariants in the algebra of polynomial functions of the containing group. Section 3 contains the fundamental results on homomorphisms from commutative algebras to the base field, culminating in Hilbert’s Nullstellensatz. Section 4 applies this to yield an important tool theorem about polynomial maps between algebraic groups, and then establishes the principal general result concerning factor groups of algebraic groups.
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