I would like to understand the following from Commutative Algebras In Which Polynomials Have Infinitely Many Roots by David E. Dobbs:
The ring of dual numbers over $\Bbb{F}_2(T)$ provides an example of a commutative finite-dimensional $\Bbb{F}_2(T)$-algebra containing infinitely many roots of $X^2+1 \in \Bbb{F}_2(T)[X]$
- What exactly is $\Bbb{F}_2(T)$? Is it another notation for $\Bbb{F}_2[T]$?
What is the ring of dual numbers over $\Bbb{F}_2(T)$? Is it $\Bbb{F}_2(T)[S]/(S^2)=\{s_1S+s_2 | s_1, s_2 \in \Bbb{F}_2(T) \}$? Is it $\Bbb{F}_2(T)/(T^2)$? Or is the variable $X$ already factored instead of $S$?
How does $X^2+1 \in \Bbb{F}_2(T)[x]$ have infinitely many roots?
I have read some threads on this site, however they are not concerned with roots of polynomials, and I tried to build small examples, but have not gained much insight.
In general, $F(\alpha)$ is the smallest field containing $F$ and $\alpha$. (This makes sense when $F$ and $\alpha$ exist in some kind of larger field.) If $\alpha$ is transcendental over $F$ then every element of $F(\alpha)$ is uniquely expressible as a rational expression in $\alpha$ (with coefficients in $F$). If $T$ is not understood to come from a larger field, it may mean a "new" transcendental, in other words a formal variable. This is the case with $\mathbb{F}_2(T)$.
Suppose $K$ is a field and $R=K[\varepsilon]/(\varepsilon^2)$ is the ring of dual numbers, an algebra over $K$. To find the roots of the polynomial $X^2+1\in R[X]$, write $\alpha=a+b\varepsilon$, plug in, set equal to $0$ and solve. You get $a=1$ and no conditions on $b$, so in particular if $K$ is infinite (as with $K=\mathbb{F}_2(T)$) there are infinitely many roots.