Question. What are the maximal subalgebras of polynomial ring $\mathbb{R}[x]$ over the field of real numbers?
6

$\begingroup$ First of all, are you sure that such maximal subalgebra exist? $\endgroup$ – GreginGre Dec 5 '18 at 7:46

$\begingroup$ Yes, for example, $A_{p_1, p_2}=\{f\in \mathbb{R}[x]\colon f(p_1)=f(p_2)\},$ where $p_1\ne p_2.$ $\endgroup$ – Vadim Sidorov Dec 5 '18 at 7:50

$\begingroup$ Also, $\mathbb{R}[x^2, x^3]$. Is there any specific reason you choose to look over $\mathbb{R}$, rather than $\mathbb{C}$? $\endgroup$ – user44191 Dec 5 '18 at 8:40

1$\begingroup$ The classification should be intepretable in terms of singularities. The first case $A_{p_1,p_2}$ consists in gluing two points in a transverse way. The second case $K[x^2,X^3]$ corresponds to a nontransverse singularity. Isolated singularities of curves are certainly wellclassified so as to fully answer the question. $\endgroup$ – YCor Dec 5 '18 at 9:06

1$\begingroup$ Is it obviously true that any maximal subalgebra is of codimension $1$ as a vector space? Because if so, then the family $K[(xa)^2, (xa)^3  b(xa)]$ (which consists of $A_{p_1, p_2}$ and translates of $K[x^2,x^3]$) exhausts all possibilities. $\endgroup$ – user44191 Dec 5 '18 at 9:35