# Notation on Hungerford Algebra Field Theory

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?

• Edited to explain why it is not a duplicate – Amphiaraos Jul 8 at 15:12
• 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). – Xander Henderson Jul 8 at 16:54