Answer to Question #351025 in Abstract Algebra for Fed

Question #351025

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


1
Expert's answer
2022-06-22T09:38:47-0400

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".


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