  For every algebraically closed field $\boldsymbol k$ of characteristic
different from $2$, we prove the following:
  (1) Generic finite dimensional (not necessarily associative) $\boldsymbol
k$-algebras of a fixed dimension, considered up to isomorphism, are
parametrized by the values of a tuple of algebraically independent over
$\boldsymbol k$ rational functions in the structure constants.
  (2) There exists an "algebraic normal form", to which the set of structure
constants of every such algebra can be uniquely transformed by means of passing
to its new basis, namely: there are two finite systems of nonconstant
polynomials on the space of structure constants, $\{f_i\}_{i\in I}$ and
$\{b_j\}_{j\in J}$, such that the ideal generated by the set $\{f_i\}_{i\in I}$
is prime and, for every tuple $c$ of structure constants satisfying the
property $b_j(c)\neq 0$ for all $j\in J$, there exists a unique new basis of
this algebra in which the tuple $c'$ of its structure constants satisfies the
property $f_i(c')=0$ for all $i\in I$.
