Hello everyone. I have been struggling with the following problem from "Algebra An Approach via Module Theory" by Adkins and Weintraub.
Let F be a field and let f be a homomorphism from F to F. Extend f to a homomorphism from F[x,y] to F[t] by setting f(x) = t^2, f(y) = t^3. Then the kernel of f is the principal ideal generated by y^2 - x^3.
Its clear that the ideal generated by y^2 - x^3 is contained in the kernel of f. To show the opposite inclusion I have resorted to writing out a general polynomial in the variables x and y and hoping to factor it (using the assumption it is in the kernel of f) so that y^2 - x^3 is a factor. This has so far been very tedious and does not seem to be the best way to go with this problem. But I am having trouble finding inspiration on this one.