- Thread starter
- #1

- Feb 29, 2012

- 342

Show that the quotient ring of a field is either the trivial one or is isomorphic to the field.

*Let $N$ be an ideal of the field $F$. Assume that $N \neq \{ 0 \}$. Consider the homomorphism $\phi: F \to F / N$ defined by $\phi(a) = a + N$. If we show that it is one-to-one and onto we are done. It is clearly surjective, thus all that is left is to show injectivity. If $a \neq b$ then we will have $a + N \neq b + N$, but this is none other than $\phi(a) \neq \phi(b)$.*

__My answer:__Thanks for all help!