1. ## ring homomorphism

phi:Q[t] -> R be defined phi(t)=rt(2) and with elements of Q mapping themselves in R. Prove that rt(8) is in Im(phi)

let h=(t^4)+(t^4)-4 show phi(h)=0

Q is rationals, R is reals.

ok i can define Im(phi) and then i say rt(8)=phi*rt(2) => phi=2..im really lost help need help pretty quick, exam revision!

2. Originally Posted by benjyboy
phi:Q[t] -> R be defined phi(t)=rt(2) and with elements of Q mapping themselves in R. Prove that rt(8) is in Im(phi)

let h=(t^4)+(t^4)-4 show phi(h)=0

Q is rationals, R is reals.

ok i can define Im(phi) and then i say rt(8)=phi*rt(2) => phi=2..im really lost help need help pretty quick, exam revision!
Part one is actually quite simple - you just have to remember that $\displaystyle \sqrt{a^n} = (\sqrt{a})^n$.

For part two you meerly substitute in $\displaystyle \sqrt{2}$ for $\displaystyle t$ as $\displaystyle \phi$ is a ring homomorphism (a ring homomorphism is a mapping from a ring to another ring such that $\displaystyle (ab+c)\phi = (a\phi)(b\phi)+c$).

I hope that helps?

Also, this would have been more suited to the "Abstract Algebra" forum as it is, well, abstract algebra...