Example 9.26.7. Let $F$ be a field and $E = F(t)$. Then $\{ t\}$ is a transcendence basis since $E = F(t)$. However, $\{ t^2\}$ is also a transcendence basis since $F(t)/F(t^2)$ is algebraic. This illustrates that while we can always decompose an extension $E/F$ into an algebraic extension $E/F'$ and a purely transcendental extension $F'/F$, this decomposition is not unique and depends on choice of transcendence basis.

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