Semidirect product of length two Witt ring and additive group

Definition

Suppose $K$ is a field of characteristic equal to a prime number $p$. The semidirect product of length two Witt ring and additive group is a three-dimensional algebraic group defined over $K$ as follows. It is the external semidirect product $A\rtimes B$ where:

• $A$ is the additive group of the truncated ring of Witt vectors over $K$ to length two.
• $B$ is the additive group of $K$.
• The action is as follows. For an element $(x_{0},x_{1})\in A$ (note that elements of $A$ are represented by pairs of elements from $K$) and an element $b\in B$ (with $b\in K$), we define:

$b\cdot (x_{0},x_{1})=(x_{0},x_{1}+bx_{0})$

where the multiplication and addition on the right in the second coordinate happens within the field $K$.

Particular cases

• In the case that $K$ is the prime field $\mathbb {F} _{p}$, the additive group of the length two Witt ring is the cyclic group of prime-square order and the semidirect product becomes semidirect product of cyclic group of prime-square order and cyclic group of prime order.