Primitive elements in $k[x]$

Question

I need to find all primitives element of $k[x]$ with the coalgebra structure: $\Delta(x)=1\otimes x+x\otimes 1$ and $\epsilon(x)=0$. If char($k$) is zero, for me it's clear that, within the basis $\{x^{n} |n \in \mathbb{N}\}$, the only possibility for an elment to be primitive is when $n=1$, so for x. But how can I be sure that for a generic polynomial $p(x)$, there won't be cancellation of terms in $\Delta(p(x))$ so that it is a primitive element too? Thank you, any hint will be appreciated !

Answer 1

Write $p(x)= a_0 +\cdots +a_nx^n$. Write $y = 1\otimes x$, and $z =x \otimes 1$. Since $\Delta$ is a ring-homomorphism, $\Delta (p (x ) ) = p(\Delta x) = p ( y + z)$. Therefore
$$ \Delta (p (x ) ) = a_0 + a_1( y + z) + a_2( y^2+ 2 yz + z^2) + \cdots.$$ For this to equal $$ (a_0 + a_1y+ a_2 y^2 + a_3 y^3+ \cdots) + (a_0 + a_1 z+ a_2 z^2 + a_3 z^3 + \cdots), $$ the coefficients on both side have to be equal, as $\{y^kz^l\}$ forms a basis for $k[y,z] $. As the characteristic of $k$ is zero... OK?