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I am working through some problems in Field Theory on Hungerford's Algebra. However I have been struggling to understand the notation in some exercises:

1.14 In the field $k(x)$, let $u = x^3/(x+1)$. Show that $K(x)$ is a simple extension of the field $K(u)$. What is $[K(x):K(u)]$?

2.11 In the extension of $\mathbb{Q}$ by $\mathbb{Q}(x)$, the intermediate field $\mathbb{Q}(x^2)$ is closed but $\mathbb{Q}(x^3)$ is not.

I understand for a field $K$ what $K(x)$ means, this is the field of rational functions with indeterminate $x$. However, I do not understand what does $K(f(x))$ means. In class we worked with examples such as $K(u)$ where $u$ is a root of a polynomial in $K[x]$.

In the linked questions the answers explain examples such as $\mathbb{Q}(\sqrt 2)$ which is the smallest field extension of $\mathbb{Q}$ that contains $\sqrt 2$. In general, for a field $K$ we have the extension $K(u)$ where $u\in F$ and $F$ is an extension of $K$.

None of the linked answers indicates what it means to have $K(f)$ where $f\in K[x]$. To be more precise is $K(f)$ a quotient $k(x)/(f)$ or is it a shorthand notation for $K(u)$ where $f(u)=0$ or is it to replace the indeterminate $x$ for $f(x)$ in the rational function?

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  • $\begingroup$ Edited to explain why it is not a duplicate $\endgroup$ – Amphiaraos Jul 8 at 15:12
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    $\begingroup$ As an aid to others, in my edition of Hungerford (the 1974 Springer edition), these exercises are V.1.15 (p. 241) and V.2.11 (p. 256). $\endgroup$ – Xander Henderson Jul 8 at 16:54

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