Answer to Question #351025 in Abstract Algebra for Fed

Question

3.6 If F is a field, prove that its only ideals are (0) and F itself.

Answer 1

Let $I$ be an ideal of a field $F$. Assume that $(0)\ne I$ and let $0\ne a\in I$. Then as $F$ is a field, there exists $a^{-1}\in F$ such that $a^{-1}a=1$. Hence $1\in I$ and $I=F$. So every non-zero ideal of $F$ is equal to $F$.