A Product Formula For Minimal Polynomials Of Rational Functions

By means of Galois theory,we give a product formula for the minimal polynomial G of rational functions {f0,f1,...,fn} in K(x1,...,xn)-K which contains n algebraically independent elements,where K is a field of characteristic zero.The main aim of this thesis is to prove the following theorem.Theorem0.1 Let rational functions {f0,f1,...,fn}(?)K(x1,...,xn)-K with f1,...,fn al-gebraically are independent over K.Let q:=[K(x1,...,xn):K(f0,f1,...,fn)]and G(t0,...,tn)be the minimal polynomial of f0,...fn.Then(ⅰ)where c ∈ K\{0},(α1(i),...,αn(i)),i=1,...,d,are all solutions of the system of equations fi(t1,...,tn)= yi,i=1,...,n,in the algebraic closure of K(y1,...,yn);and D ∈K[y1,...,yn]is the unique minimal denominator(up to a constant factor in K\{0})of the product(?)(ⅱ)The degrees for yi of G is degyi(G)=di/q,where di is the number of solutions of the system of equations fj(t1,...,tn)-yj=0,j=0,1,...,i-1,i+1,...,n,in K(y1,...,yn).If di>0,then di=[K(x1,...,xn):K(f0,...,fi-1,fi+i,...,fn)];(ⅲ)The total degree of G satisfieswhere fi=vi/wi,vi,wi∈K[x1,...,xn]and deg(fi):=max{deg(vi),deg(wi)},vi,wi∈K[x1,...,xn],i=0,...,n.