What are the primitive elements in a polynomial Hopf algebra with primitive indeterminates? Asked on math.stackexchange https://math.stackexchange.com/questions/2510606/what-are-the-primitive-elements-in-a-polynomial-hopf-algebra-with-primitive-inde but didn't get response (in fact got a negative vote without any comment), so trying here. 
Is it true that in any polynomial Hopf algebra $K[X1,X2,...]$ over a field $K$ with indeterminates primitive, the primitive elements are precisely the linear homogeneous polynomials? (Perhaps with some additional assumptions like characteristic of $K$ is $0$?). If so, could someone kindly give me a reference? A paper I am reading says (without citation) that it is well-known. 
 A: No, in general the claim is not true: 
To see why, consider a field $k$ of characteristic $p$ and take the polynomial hopf algebra $k[x]$ (in a single variable). Then $x$ is primitive and so is $x^p$: 
$$
\Delta(x^p)=1\otimes x^p+x^p\otimes 1
$$ 
(because in characteristic $p$: $\binom{p}{i}=0$, for $1\leq i\leq p-1 \ $).  
However, in characteristic zero, $k[x]$ is generated by its primitive elements,  which are in fact the homogeneous linear polynomials, in the sense that: $P(k[x])=kx$, as sets,and $k[x]\cong U\big(P(k[x])\big)\cong T(kx)$  as Hopf algebras.
($P(\cdot)$ denotes the Lie algebra of the primitives, $U(\cdot)$ stands for  the universal enveloping algebra of $P(\cdot)$ and $T(.)$ the tensor or symmetric algebra of the one dimensional vector space $kx$).
  Since you are also asking for some reference, the last statement (on characteristic zero) can be found cited explicitly (apart from the celebrated Milnor-Moore paper already cited at the comments to the OP) at the following sources: 


*

*Dascalescu's book, "Hopf algebras. an introduction", p.166, example 8 

*Montgomery's book, "Hopf algebras and their actions on rings", p.82, (see the discussion right after the example 5.6.8)

*Abe's book, "Hopf algebras", p.61, example 2.6  

