# Dickson invariant

A construction used in the study of quadratic forms over fields of characteristic 2, which allows one, in particular, to introduce analogues of the special orthogonal group over such fields. In fact, a Dickson invariant is an element $D(u)$ of a field $k$ of characteristic 2 associated to any similarity $u$ of a countable-dimensional vector space $E$ over $k$ with respect to the symmetric bilinear form $f$ associated with a non-degenerate quadratic form $Q$ on $E$. Introduced by L.E. Dickson .

By virtue of the condition imposed on the characteristic of the field, the form $f$ is alternating and there exists a basis $e_1,\dotsc,e_{2s}$ in $E$ for which

$$f(e_i,e_j)=f(e_{s+i},e_{s+j})=0,$$

$$f(e_i,e_{s+j})=\delta_{ij},$$

for $1\leq i\leq s$, $1\leq j\leq s$ (cf. Witt decomposition). Let

$$f(u(x),u(y))=\alpha(u)f(x,y)$$

for any vectors $x$ and $y$ from $E$, and let, for each $i=1,\dotsc,s$,

$$u(e_i)=\sum_{j=1}^sa_{ij}e_j+\sum_{j=1}^sb_{ij}e_{s+j},$$

$$u(e_{s+i})=\sum_{j=1}^sc_{ij}e_j+\sum_{j=1}^sd_{ij}e_{s+j}.$$

Then the following element from $k$:

$$D(u)=\sum_{i,j}(Q(e_i)a_{ij}c_{ij}+Q(e_{s+i})b_{ij}d_{ij}+b_{ij}c_{ij})$$

is called the Dickson invariant of the similarity $u$ with respect to the basis $e_1,\dotsc,e_{2s}$. For $u$ to be a similarity with respect to $Q$ with similarity coefficient $\alpha(u)$ (i.e. $Q(u(x))=\alpha(u)Q(x)$ for any vector $x\in E$) it is necessary and sufficient that $D(u)=0$ or that $D(u)=\alpha(u)$. Similarities $u$ with respect to $Q$ for which $D(u)=0$ are called direct similarities. The direct similarities form a normal subgroup of index 2 in the group of all similarities with respect to $Q$.

If $Q_1$ is the form defined by $Q_1(x)=Q(u(x))$ for any vector $x\in E$, and if $\Delta(Q)$ and $\Delta(Q_1)$ are the pseudo-discriminants of these forms with respect to the basis $e_1,\dotsc,e_{2s}$, i.e.

$$\Delta(Q)=Q(e_1)Q(e_{s+1})+\dotsb+Q(e_s)Q(e_{2s}),$$

$$\Delta(Q_1)=Q_1(e_1)Q_1(e_{s+1})+\dotsb+Q_1(e_s)Q_1(e_{2s}),$$

then

$$\Delta(Q_1)=(\alpha(u))^2\Delta(Q)+(D(u))^2+\alpha(u)D(u).$$

How to Cite This Entry:
Dickson invariant. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dickson_invariant&oldid=53616
This article was adapted from an original article by V.L. Popov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article