Recent content by ivos

I can prove 1), 2) without linear algebra: Let f,g be irreducible polynomials over field F. Let u be root of f, v be root of g. Assume, that F(u)=F(v). Then deg f = deg g. Proof: Its well known (it can be proven by basic polynomial algebra) that any element of F(u), in particular v...

f(x)=k.g(x) is not valid (in generally) F=Q, f(x)=x^2-2,g(x)=x^2-2*x-1, u=sqrt(2), v=sqrt(2)+1 Then Q(u)=Q(v), but f(x) is not equals k.g(x)...

Let F be a field of characteristic 0. Let f,g be irreducible polynomials over F. Let u be root of f, v be root of g; u,v are elements of field extension K/F. Let F(u)=F(v). Prove (with using basic polynomial theory only, without using linear algebra and vector spaces): 1) deg f = deg g (deg f...