Additive group of ring of Witt vectors inherits algebraic group structure
For ring of Witt vectors
Suppose is a field and is a prime number. Suppose denotes the ring of Witt vectors over for the prime . Then, the additive group of naturally inherits the structure of an algebraic group over , with the algebraic group structure arising from the fact that the operations are defined by formulas with coordinates in , and that these formulas work to give a group in any field extension of .
For ring of Witt vectors truncated to finite length
Suppose is a field, is a prime number, and is a natural number. Consider the ring of truncated Witt vectors of length , obtained by projecting the ring of Witt vectors to the first coordinates. The additive group of this ring is an algebraic group over .