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Tutorials for Algebra-III

Abhay Soman

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Worksheet Tutorial-7

Following exercises are taken from Section 14.1 of the book Abstract Algebra by Dummit and Foote.
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1.

Let \(X\) be a nonempty subset of a field \(K\text{,}\) and let \(K\) be generated over a field \(F\) by \(X\text{,}\) i.e., \(K=F(X)\text{.}\) Show that any automorphism \(\sigma\) of \(K\) fixing \(F\) is uniquely determined by the action \(\sigma\) on \(X\text{.}\)
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2.

Let \(G\leq\Gal{K}{F}\) be a subgroup of the Galois group of the extension \(K/F\text{.}\) Assume that \(G\) is generated by \(\sigma_1,\sigma_2,\ldots,\sigma_n\text{.}\) Show that the subfield \(E/F\) is fixed by \(G\) if and only if \(E/F\) is fixed by \(\sigma_1,\sigma_2,\ldots,\sigma_n\text{.}\)
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3.

Determine all automorphisms of \(\Q\left(\sqrt[4]{2}\right)\) fixing \(\Q\left(\sqrt{2}\right)\text{.}\)
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4.

Show that \(\Aut{\R}{\Q}=\{\unit_\R\}\) by proving the following.
  1. Show that any \(\sigma\in\Aut{\R}{\Q}\) takes squares to squares and a positive real number to a positive real number. Thus, show that if \(a<b\) then, \(\sigma a<\sigma b\text{.}\)
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  2. Show that \(\sigma\in\Aut{\R}{\Q}\) is a continuous map by proving that the following.
    For \(m\in\N\text{,}\) if \(\tfrac{-1}{m}<a-b<\tfrac{1}{m}\) then, \(\tfrac{-1}{m}<\sigma a-\sigma b<\tfrac{1}{m}\text{.}\)
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  3. Show that a continous real-valued map on \(\R\) which is identity on \(\Q\) is identity on \(\R\text{.}\)
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5.

Show that the automorphisms of the rational function field \(F(t)\) fixing \(F\) are precisely fractional linear transformations detrmined by \(t\mapsto at+b\big/ct+d\text{,}\) where \(a,b,c,d\in F\) with \(ad-bc\neq 0\text{.}\)
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Determine the fixed field of the automorphism of \(F(t)\) given by
\begin{equation*} \frac{f(t)}{g(t)}\mapsto \frac{f(1+t)}{g(1+t)}\text{.} \end{equation*}
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6.

Let \(K/F\) be a field extension and \(\varphi\colon K\to K'\) an isomorphism of fields. Let \(F'=\varphi(F)\text{.}\) Show that the map \(\Inn{\varphi}\colon\Aut{K}{F}\to\Aut{K'}{F'}\) defined by \(\sigma\mapsto\varphi\circ\sigma\circ\varphi^{-1}\) is a group isomorphism.
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