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And let $\phi$ be the homomorphism $K \rightarrow K$ which sends $t \rightarrow 2/t$.

a. Show that $\phi$ is an automorphism of K of order 2.

b. Show that the fixed field of $G=\{1,\phi\}$ is $F=\mathbb{Q}(t+2/t)$.

c. What is the minimal polynomial of $t$ over $F$.

First of all I don’t understand what is meant by field of rational functions in transcendental element?? Can I get Somen hint for a? I think then b will follow just by definition . And what about c?

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The field of rational functions in transcendental element, $\mathbb{Q}(t)$, is defined to be the field of fractions of the polynomial ring $\mathbb{Q}[t]$. You can think of it as elements of the form $$\frac{p(t)}{q(t)}=\frac{a_0+a_1t+\ldots+a_nt^n}{b_0+b_1t+\ldots+b_mt^m}$$ where all of the $a_i,b_i\in\mathbb{Q}$, $b_i\not=0$ for some $i$.

To show that it's an automorphism of order $2$, you need to show that it's a function from $\mathbb{Q}(t)$ to $\mathbb{Q}(t)$ and that it has order $2$ (which means that composing it with itself you get the identity function, and it's not the identity function).

For c), you can view $\mathbb{Q}(t)$ as an infinite-dimensional vector space over $\mathbb{Q}$. In this context, $\phi$ is a map of $\mathbb{Q}$-vector spaces, and since it has finite order, it has a minimal polynomial. What is that minimal polynomial?

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