Question

Given $a_1,b_1,c_1$ in $F_2[t]$ with $\gcd(a_1,b_1,c_1)=1$ it is known that there exists an element $g$ of $SL(3, F_2[t])$ (by explicit construction) such that $g$ has first line $$ [a_1^2, b_1,c_1], $$ in particular this works when $a_1^2$ is in the ideal $(b_1,c_1)$ generated by $a_1$ and $b_1$ in $F_2[t].$

Question: It is possible to extend in the above manner the following line $L$ to an element of $SL(3, F_2[t])$

$$ L =[1+a,b,c] $$ where $a,b,c \in F_2[t]$ with $\gcd(1+a,b,c)=1$ and with $$ a^2 \in (b,c). $$

The case when $\gcd(b,c)=1$ is trivial since $1+a = M^2+ t N^2$ and $tN^2 \in (b,c)$ in these case.

The question (for a general characteristic $2$ ring R) appears in page $14$ of

www.math.psu.edu/oldColloquium/Ravi2.pdf

while the explicit construction described above appears in page $12.$

Answer 1

Okay, suppose $K$ is a field and you have three elements $a,b,c\in K[t]$ with $gcd(a,b,c)=1$. I will show that the row $(a,b,c)$ can always be extended to a matrix in $SL(3,K[t])$. Suppose that at least one of $b$ and $c$ is nonzero (otherwise $a\in K$ and extending the row is trivial).

Let $g:=gcd(b,c)$ and choose $x,y\in K[x]$ such that $xb+yc=g$. Since $gcd(a,g)$ must be one, we find $r,s\in K[x]$ such that $ra+sg=1$. Now the following $3\times 3$-matrix has determinant $1$ and extends the row $(a,b,c)$: $$ \left[\begin{array}{ccc}a&b&c\newline -s& \frac{rb}{g} & \frac{rc}{g} \newline 0 & -y & x\ \end{array}\right] $$ It is an easy computation that this matrix has determinant one (unless I mistyped something). Note that the occurring fractions lie in $K[x]$, since $g$ divides both $b$ and $c$ by definition.

Answer 2

Let $R$ be an arbitrary ring, and $(x,y,z)$ a unimodular row. View this row as a linear map from $R$ to $R^3$. Then the unimodularity implies that this map splits, so its cokernel (call it $M$) is projective.

Now suppose that $M$ is free, and that free modules over $R$ have well-defined ranks (e.g. if $R$ is commutative). Then $M$ is isomorphic to $R^2$. A complementary splitting $R^2\rightarrow R^3$ tells you how to complete your matrix.

In particular, this works whenever all projective $R$-modules are free, which is certainly the case for any PID and in particular for $F_2[t]$.