Unipotent linear algebraic group structure of additive group of ring of Witt vectors of length two

This article describes the structure of the additive group of the truncated ring of Witt vectors to length two over a ring $$R$$ in characteristic $$p$$ as a linear algebraic group of degree $$p$$ over $$R$$.

Explicitly, the map is:

$$(X_0,X_1) \mapsto M(X_0,X_1)$$

where $$M$$ is a $$(p+1)\times (p+1)$$ matrix defined as follows:


 * All entries below the diagonal are zero.
 * All entries on the diagonal are 1.
 * The top right entry is $$X_1$$.
 * The other entries are given as follows:

Case of the prime two
In this case, the mapping is as follows:

$$(X_0,X_1) \mapsto \begin{pmatrix} 1 & X_0 & X_1 \\ 0 & 1 & X_0 \\ 0 & 0 & 1 \\\end{pmatrix}$$

In particular, if we consider these matrices over field:F2 and let $$X_0,X_1$$ vary over the elements of the field, the image of the mapping, a a multiplicative matrix group, is isomorphic to the additive group cyclic group:Z4.

Case of prime three
$$(X_0,X_1) \mapsto \begin{pmatrix} 1 & X_0 & X_0^2 & X_1 \\ 0 & 1 & 2X_0 & X_0^2 \\ 0 & 0 & 1 & X_0 \\ 0 & 0 & 0 & 1 \\\end{pmatrix}$$

In particular, if we consider these matrices over field:F3 and let $$X_0,X_1$$ vary over the elements of the field, the image of the mapping, a a multiplicative matrix group, is isomorphic to the additive group cyclic group:Z9.