Suppose fx and gx are two polynomials in Qx Suppose that the

Suppose f(x) and g(x) are two polynomials in Q[x]. Suppose that the splitting field of f(x) is of dimension n over Q, and the splitting field. Of g(x) is of dimension m over Q. Prove that the splitting field of f(x). g(x) has dimension no more than n. m.

Solution

let f(x) and g(x) be two polynomials inQ[x].The splitting feild of f(x) is dim n and the splitting field over g(x) is of dim m.we have

deg f(x)=n and deg g(x)=m from this gcd(f(x),g(x))= 1 this follows that f(x) divides g(x) we have the degree of minimal npolynomial of f(x) is divisible by g(x).considering the former polynomial divides the latter we have the two polynomials are equal and this infers that the splitting field f(x).g(x) has no dimension then n.m.