Substitution Homomorphism from F[x,y] to F[t]

**Kopeck** · July 15th 2011, 07:12 PM

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.

**NonCommAlg** · July 15th 2011, 10:14 PM

Originally Posted by Kopeck

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.

let $I = \langle y^2-x^3 \rangle$ and $u \in \ker f$ . then, since $y^2 \equiv x^3 \mod I,$ we have $u \equiv p(x) + yq(x) \mod I$ , for some $p(x), q(x) \in F[x]$ . since $I \subseteq \ker f$ and $u \in \ker f,$ we have $p(x) + yq(x) \in \ker f,$ i.e. $p(t^2)+t^3q(t^2)=0$ . but all powers of $t$ in $p(t^2)$ are even numbers and all powers of $t$ in $t^3q(t^2)$ are odd numbers. thus $p(t^2)+t^3q(t^2)=0$ implies $p = q = 0$ and so $u \equiv 0 \mod I,$ i.e. $u \in I. \ \Box$

**Kopeck** · July 16th 2011, 05:19 PM

Thank you very much for the help NonCommAlg. This is a very cool technique.

Thread: Substitution Homomorphism from F[x,y] to F[t]

LinkBack

Thread Tools

Display

Substitution Homomorphism from F[x,y] to F[t]

Re: Substitution Homomorphism from F[x,y] to F[t]

Re: Substitution Homomorphism from F[x,y] to F[t]

Similar Math Help Forum Discussions

Proof that homomorphism of an inverse equals the inverse of the homomorphism

homomorphism help

homomorphism

Homomorphism

Homomorphism

Search Tags