Take the tour ×  Let $A$ be a commutative ring and $n \in
\mathbb{N}$. What is the minimal number $e_A(n)$ of generators
of the $A$-algebra $A^n$? Here is what I already know (I can add
proofs if necessary) from a little joint work:    1. We have
$e_A(n)=0$ for $n=0,1$, therefore let's exclude these trivial
cases. We have $e_A(2) \leq 1$ since $A^2$ is generated by
$(1,0)$.    2. We have $e_A(mn) \leq e_A(m) + e_A(n)$ since
$A^{mn} \simeq A^m \otimes_A A^n$ as $A$-algebras.    3. The
function $e_A : \mathbb{N} \to \mathbb{N}$ is non-decreasing.
4. When there is a homomorphism $A \to B$, then $e_B \leq e_A$
holds pointwise. In particular we have $e_A = e_B$ when $B$ is
an $A$-algebra with a section. This happens, for example, when
$B$ is a polynomial ring over $A$.    5. We have $e_A(n) \leq
\lceil \log_2(n) \rceil$ (use 1,2,3).    6. If $A$ has elements
$\alpha_1,\dotsc,\alpha_n$ such that $\alpha_i - \alpha_j \in
A^*$ for $i \neq j$, then $e_A(n)=1$ (use Vandermonde). In
particular: If $K$ is a field with