If are roots of polynomials over a field of degrees respectively, and is a polynomial over with respect to n variables , then construct a polynomial over which has the root .

For constructing this, set a polynomial of the form

where and the coefficients are yet to be determined. Divide by , divide the remainder by and, iterating this process, finally get a remainder . All coefficients of can be written as linear combinations of ’s. Any monomial of the polynomial is of the form and thus the polynomial consists of less monomials than , that is, the number of all coefficients of is less than .The number of all coefficients of is . Hence we can choose non-trivial (i.e. some is not zero) ’s such that all coefficients of are zero and obtain , which has the root .

Example1.

Example2.