Let $F/K$ be a field extension and $a\in F$ such that $[K(a):K]$ is odd. Prove that $K\left(a\right)=K\left(a^{2}\right)$ and give counterexample if $[K(a):K]$ is even.
Asked
Active
Viewed 100 times
1
-
1Hint: $K \subset K(a^2) \subset K(a)$. Use the degree formula. – Daniel Fischer Nov 14 '13 at 10:52
-
5It seem to be answered here http://math.stackexchange.com/questions/77769/equal-simple-field-extensions But I want a counterexample if $[K(a):K]$ is even – Truong Nov 14 '13 at 10:55
-
1@chuyenvien94 Consider $F=\mathbb{R}, K=\mathbb{Q}, a=\sqrt{2}$. Claeraly, $\mathbb{Q}(\sqrt{2})\not= \mathbb{Q}((\sqrt{2})^2)$ – Amr Nov 14 '13 at 10:58
-
Oh, of course. Thank you. – Truong Nov 14 '13 at 10:59
-
1@chuyenvien94 Would you answer you own question with the comments you and other have made? – BIS HD Nov 18 '13 at 10:12