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Tutorials for Algebra-III

Abhay Soman

Worksheet Tutorial-9

Following exercises are taken from Section 14.2 of the book Abstract Algebra by Dummit and Foote.
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1.

Suppose that \(K/F\) is a Galois extension of degree \(p^n\) for some prime number \(p\) and a natural number \(n\text{.}\) Show that there are Galois extensions of \(F\) contained in \(K\) of degrees \(p\) and \(p^{n-1}\text{.}\)
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2.

Give an example of fields \(F_1,F_2,F_3\) with \(\Q\subset F_1\subset F_2\subset F_3\) and \([F_3:\Q]=8\) and each field is Galois over all its subfields with the exception that \(F_2/\Q\) is not Galois.
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3.

Determine the Galois group of the splitting field of \(x^4-14x^2+9\) over \(\Q\text{.}\)
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4.

Show that \(\Q\left(\sqrt{2+\sqrt{2}}\right)/\Q\) is Galois with a cyclic Galois group.
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5.

  1. Show that the polynomial \(p(x)=x^4-2x^2-2\) is irreducible over \(\Q\text{.}\)
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  2. Show that following are roots of \(p(x)\text{.}\)
    \begin{align*} \alpha_1=\sqrt{1+\sqrt{3}} \amp\qquad\alpha_2=-\sqrt{1+\sqrt{3}} \\ \alpha_3=\sqrt{1-\sqrt{3}} \amp\qquad\alpha_4=-\sqrt{1-\sqrt{3}} \end{align*}
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  3. Let \(K_i=\Q(\alpha_i)\) for \(i=1,2\) and \(F=K_1\cap K_2\text{.}\) Show that \(K_1\neq K_2\) and that \(F=\Q(\sqrt{3})\text{.}\)
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  4. Show that \(K_i/F\) are Galois for \(i=1,2\text{.}\) Determine \(\Gal{K_1K_2}{F}\) and all the subfields of \(K_1K_2/F\) explicitly.
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  5. Prove that the splitting field of \(p(x)\) over \(\Q\) is of degree \(8\) and its Galois group is isomorphic to a dihedral group.
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