Combinatorial problems can often be translated into
polynomial equations over the Galois field $\mathbb{GF}_2$.
A Sudoku puzzle that is one of the most popular puzzles in a world today
is among such instances.
In a naive and natural way, it can be translated into a system of
polynomial equations over the Galois field $\mathbb{GF}_2$ with 729
variables. Unless we have many initial conditions, however, it is
impossible to solve the equation using a general propose algorithm
to solve polynomial equations over $\mathbb{GF}_2$.

We propose an algorithm to solve Sudoku puzzles by computations
of Boolean Gr\"obner bases for a Boolean ring of the power set
$P(\{ 1,2,\ldots,9\})$ of $\{ 1,2,\ldots,9\}$.

For a given Sudoku puzzle, our algorithm computes every minimal
polynomials. These polynomials can be considered as
a kind of parametric solution of the puzzle.
Using these polynomials, we can not only solve the puzzle but
also detect its level of difficulty.