Assignment:
Suppose F, E are fields and F is a subring of E. Suppose g is an element of E and is algebraic over F, that is p(g)=0 for some nonzero polynomial p(x) in F[x]. Then there must be a nonzero polynomial m(x) smallest degree among those nonzero polynomials in F[x] with g as a root. Prove m(x) cannot be factored as a(x)b(x) for polynomials a(x), b(x) of smaller degree in F[x].
The suppose q(x) in F[x] with q(g)=0. Show q(x) is an element of m(x)F[x] (so ker(Eg:F[x]-->E) = m(x)F[x])
Then show that the subring F[g] of E is isomorphic to the quotient ring F[x]/m(x)F[x].
Provide complete and step by step solution for the question and show calculations and use formulas.