Suppose $SL_2(\mathbf{C})$ acts on the space of quadratics $aX^2+2bXY+cY^2$ by $X\to \alpha X+\beta Y, Y\to\gamma X+\delta Y$, where $\alpha,\beta,\gamma,\delta$ consists a matrix in $SL_2(\mathbf{C})$. How to show the invariant polynomials in $a,b,c$ are polynomials in $ac-b^2$?
Does this hold for $SL_2(\mathbf{R})$?
Algebra of $SL_2(\mathbb C)$-invariants of a binary form of degree 2 is isomorphic to intersection of kernels of two derivations of the ring $\mathbb C[a,b,c]$ which are corresponding to the two one-parametric subgroups of the $SL_2(C).$ So, the transcendence degree of the algebra invariants is equal to $3-(1+1)=1.$ One invariant we already have: $ac-b^2$. Thus it generates the whole algebra of invariants.