Is there a finite-dimensional algebra over $\mathbb C$ with no zero divisors?

Question

Every non-trivial finite dimensional associative algebra over an algebraically closed field has zero divisors. Is there a non-trivial finite dimensional non-associative algebra over $\mathbb C$ or some other algebraically closed field with no zero divisors?

Answer 1

It suffices to check this for unital algebras. Let $K$ be an algebraically closed field. For any finite dimensional algebra $A$ over $K$ we consider an element $x\in A\setminus K$ and the linear operator $L_x(a)=xa$. Denote by $L$ the matrix of $L_x$ with respect to some basis. The polynomial $p(\lambda)=det(\lambda I_n-L)$ has a root $r$, since we are working over an algebraically closed field.

This shows that the (linear) operator $D=\lambda I_n-L_x$ has non trivial kernel. Therefore $\lambda v=xv$ for some $0\neq v\in A$ and since $x\notin K$ we have $\lambda-x\neq0$ and $(\lambda-x)v=0$ hence $A$ has zero divisors.