Gröbner Bases

There are many different ways to look at the theory of Gröbner bases. In the context of classical algebra, the natural point of view is as follows. Suppose first we are given univariate polynomials f, g₁, …, g _m over a field, and we wish to decide whether f is in the ideal generated by the g _i According to the results of Section 2.2, the thing to do is to compute the gcd g of the g _i and then perform long division of f by g. The polynomial / will lie in the ideal in question if and only if the remainder of this division equals zero. Moreover, if this is the case, then one also obtains a polynomial q that satisfies f = qg, namely, the quotient of the division, which equals the sum of the monomial multipliers that were used in the individual steps of the division.