# Maximal subalgebras in polynomial ring $\mathbb{R}[x]$ over the field $\mathbb{R}$ of real numbers

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

• First of all, are you sure that such maximal subalgebra exist? – GreginGre Dec 5 '18 at 7:46
• Yes, for example, $A_{p_1, p_2}=\{f\in \mathbb{R}[x]\colon f(p_1)=f(p_2)\},$ where $p_1\ne p_2.$ – Vadim Sidorov Dec 5 '18 at 7:50
• Also, $\mathbb{R}[x^2, x^3]$. Is there any specific reason you choose to look over $\mathbb{R}$, rather than $\mathbb{C}$? – user44191 Dec 5 '18 at 8:40
• 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 non-transverse singularity. Isolated singularities of curves are certainly well-classified so as to fully answer the question. – YCor Dec 5 '18 at 9:06
• Is it obviously true that any maximal subalgebra is of codimension $1$ as a vector space? Because if so, then the family $K[(x-a)^2, (x-a)^3 - b(x-a)]$ (which consists of $A_{p_1, p_2}$ and translates of $K[x^2,x^3]$) exhausts all possibilities. – user44191 Dec 5 '18 at 9:35