I'm learning Invariant Theory of Finite Groups and saw this problem:
"What is $S^G$ where $S := \mathbb{F}_p[x, y]$, and $G$ is the subgroup of $GL_2(\mathbb{F}_p)$ generated by $\begin{pmatrix} 1&1\\0&1 \end{pmatrix}$"
Since the characteristic of $S$ and $|G|$ are both $p$, this is clearly the modular case. This is weird because up to this problem in the note I'm reading, the materials only cover the nonmodular case.
There's no Noether bound here so I don't really know what to do. When I'm doing the normal way, even when I consider the invariants of degree $2$, there are two cases depending on $p$:
If $p=2$, $x^2+xy$ and $y^2$ generate the invariants in degree $2$.
If $p\neq 2,$ $y^2$ generate the invariants in degree $2$.
The point is, I suppose I can generalize this for low degrees, but without a bound, I won't get anything out of this at the end.