Mathematics Asked on January 25, 2021
How would I prove this?
I know that I must show $f(a)=f(b) Rightarrow a = b$
I also know I must use the definition of homomorphism, ie:
$f(a+b)=f(a)+f(b)$
$f(ab)=f(a)f(b)$
$f(1)=1$
I am assuming that a contradiction would be a good approach to solve this, but not quite sure on specifics.
Suppose $f(a) = f(b)$, then $f(a-b) = 0 = f(0)$. If $u = (a-b) ne 0$, then $f(u)f(u^{-1}) = f(1) = 1$, but that means that $0 f(u^{-1}) = 1$, which is impossible. Hence $a - b = 0$ and $a = b$.
Correct answer by John Hughes on January 25, 2021
The kernel of $f$ must be equal to ${0}$. Suppose that $f(a)=0$ for $ane0$; then for all $b$ one has $f(ab)=f(a)f(b)=0$. Since, for $ane0$ the function $arightarrow ab$ is onto in a field we have $f=0$ which is not possible because $f(1)=1$. It follows the kernel of $f$ is ${0}$.
Answered by Piquito on January 25, 2021
To prove it is an injective we must show that if $f(a) = f(b)$ then $a = b$.
Now suppose $f(a) = f(b)$ but $a ne b$, then we have that $f(a) - f(b) = 0$ and $u = a - b ne 0$.
So $f(u u^{-1}) = f(u)f(u^{-1}) = 1$, but $f(u) = f(a - b) = f(a) - f(b) = 0$. We have that $f(u)f(u^{-1}) = 1 = 0 f(u^{-1})$ in which for field is impossible. Thus $a = b$.
Answered by Ari Royce on January 25, 2021
A field homomorphism must in particular be a ring homomorphism, so its kernel is an ideal. The only ideals of a field are the zero ideal and the field itself.
Answered by ncmathsadist on January 25, 2021
Get help from others!
Recent Answers
Recent Questions
© 2024 TransWikia.com. All rights reserved. Sites we Love: PCI Database, UKBizDB, Menu Kuliner, Sharing RPP