Fix a monomial order on \(k[x_1,\ldots,x_n]\). A Gröbner basis of an ideal \(I\) is a finite subset of \(I\) whose leading terms generate the initial ideal \(\operatorname{in}(I)\), and Buchberger’s algorithm computes one from any finite generating set. The remainder of \(f\) on division by a Gröbner basis is unique, and \(f\in I\) if and only if that remainder is \(0\). For the lexicographic order with \(x_1>\cdots>x_n\) and a Gröbner basis \(G\) of \(I\), \(G\cap k[x_{j+1},\ldots,x_n]\) is a Gröbner basis of the elimination ideal \(I\cap k[x_{j+1},\ldots,x_n]\), which solves a polynomial system one variable at a time. For any field, \(f\in\sqrt{I}\) if and only if \(1\in I+(1-yf)\subseteq k[x_1,\ldots,x_n,y]\) (the Rabinowitsch trick). For \(k\) algebraically closed, Hilbert’s Nullstellensatz states that \(f\) vanishes on \(V(I)\) if and only if \(f\in\sqrt{I}\), and \(V(I)\) is finite if and only if \(\dim_kk[x_1,\ldots,x_n]/I<\infty\); the monomials not in \(\operatorname{in}(I)\) form a \(k\)-basis of \(k[x_1,\ldots,x_n]/I\).