1.
Show that the field \(F\left(a_1,\ldots,a_n\right)\) is the compositum of fields \(F(a_i)\text{.}\)
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Worksheet Tutorial-3
Following exercises are taken from Section 13.2 of the book Abstract Algebra by Dummit and Foote.
2.
Show that \(\Q\left(\sqrt{2},\sqrt{3}\right)=\Q\left(\sqrt{2}+\sqrt{3}\right)\text{.}\) Find the minimal polynomial of \(\sqrt{2}+\sqrt{3}\) over \(\Q\text{.}\)
3.
Let \({\rm char}(F)\neq 2\text{.}\) Let \(D_1,D_2\in F\setminus F^2\text{.}\) Show the following.
\begin{equation*}
\left[F\left(\sqrt{D_1},\sqrt{D_2}\right):F\right]=\begin{cases}
4\amp\text{if }D_1D_2\not\in F^2\\
2\amp\text{otherwise}
\end{cases}
\end{equation*}
When \(\left[F\left(\sqrt{D_1},\sqrt{D_2}\right):F\right]=4\) the field extension \(F\left(\sqrt{D_1},\sqrt{D_2}\right)/F\) is called a biquadratic extension.
4.
Let \({\rm char}(F)\neq 2\text{.}\) Suppose that \(a,b\in F\) and that \(b\not\in F^2\text{.}\) Show that \(\sqrt{a+\sqrt{b}}=\sqrt{m}+\sqrt{n}\) for some \(m,n\in F\) if and only if \(a^2-b\in F^2\text{.}\) Use this to determine when the field extension \(\Q\left(\sqrt{a+\sqrt{b}}\right)\big/\Q\) has degree \(4\text{.}\)
5.
Suppose that \(K=\Q(a_1,\ldots,a_n)\) is such that \(a_i^2\in\Q\) for \(i=1,\ldots,n\text{.}\) Show that the real cube root of \(2\) does not belong to \(K\text{.}\)
6.
7.
Let \(k\) be a field and let \(k(x)\) be the field of rational functions in one variable over \(k\text{.}\) Let \(t=P(x)/Q(x)\text{,}\) where \(P(x),Q(x)\in k[x]\text{,}\) \(Q(x)\neq 0\text{,}\) and \((P,Q)=1\text{.}\) Assume that \(t\not\in k\text{.}\) Compute \([k(x):k(t)]\) using the following steps.
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Show that \(R(s)= P(s)-tQ(s)\in k(t)[s]\) is irreducible over \(k(t)\) and that \(x\) is a root of \(R(s)\text{.}\)
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Show that \(\deg_s R(s)=\max\{\deg_xP(x),\deg_xQ(x)\}\text{.}\)
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Show that \([k(x):k(t)]=\max\{\deg_xP(x),\deg_xQ(x)\}\text{.}\)
8.
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Show that the following map is \(F\)-linear.\begin{equation*} \ell_\alpha\colon K\to K\quad\text{given by}\quad x\mapsto\alpha x \end{equation*}
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Suppose that \(A\in M_n(F)\) be a matrix of the linear map \(\ell_\alpha\text{.}\) Show that \(\alpha\) is a root of the characteristic polynomial of \(A\text{.}\) This gives a procedure to determine a polynomial of degree \(n\) satisfying an element \(\alpha\) in an extension of degree \(n\) over \(F\)
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Using the procedure outlined in the above two sub-questions, find a monic polynomial of degree \(3\) satisfied by \(\sqrt[3]{2}\) (resp. \(1+\sqrt[3]{2}+\sqrt[3]{4}\)).
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Show that \(K\) is isomorphic to a subfield of \(M_n(F)\text{.}\) So the ring \(M_n(F)\) contains an isomorphic copy of every field extension of \(F\) of degree at most \(n\text{.}\)
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Let \(D\in\Q\setminus\Q^2\) and let \(K=\Q\left(\sqrt{D}\right)\text{.}\) Let \(\alpha=a+b\sqrt{D}\in K\text{.}\)
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Show that the matrix of \(\ell_\alpha\) with respect to the basis \(\{1,\sqrt{D}\}\) is the following.\begin{equation*} \begin{pmatrix} a\amp b D\\b\amp a \end{pmatrix} \end{equation*}
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Prove directly that the following is an isomorphism of \(K\) with a subfield of \(M_2(\Q)\text{.}\)\begin{equation*} a+b\sqrt{D}\mapsto\begin{pmatrix} a\amp bD\\b\amp a\end{pmatrix} \end{equation*}
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