Compute each of the following Galois groups. Which of these field extensions are normal field extensions? If the extension is not normal, find a normal extension of \({\mathbb Q}\) in which the extension field is contained.
then \(G(\gf(729)/ \gf(9)) \cong {\mathbb Z}_3\text{.}\) A generator for \(G(\gf(729)/ \gf(9))\) is \(\sigma\text{,}\) where \(\sigma_{3^6}( \alpha) = \alpha^{3^6} = \alpha^{729}\) for \(\alpha \in \gf(729)\text{.}\)
Let \(E\) be the splitting field of a cubic polynomial in \(F[x]\text{.}\) Show that \([E:F]\) is less than or equal to \(6\) and is divisible by \(3\text{.}\) Since \(G(E/F)\) is a subgroup of \(S_3\) whose order is divisible by \(3\text{,}\) conclude that this group must be isomorphic to \({\mathbb Z}_3\) or \(S_3\text{.}\)
Let \(F \subset K \subset E\) be fields. If \(E\) is a normal extension of \(F\text{,}\) show that \(E\) must also be a normal extension of \(K\text{.}\)
Let \(p\) be a prime and \({\mathbb Z}_p(t)\) be the field of rational functions over \({\mathbb Z}_p\text{.}\) Prove that \(f(x) = x^p - t\) is an irreducible polynomial in \({\mathbb Z}_p(t)[x]\text{.}\) Show that \(f(x)\) is not separable.
Let \(E\) be an extension field of \(F\text{.}\) Suppose that \(K\) and \(L\) are two intermediate fields. If there exists an element \(\sigma \in G(E/F)\) such that \(\sigma(K) = L\text{,}\) then \(K\) and \(L\) are said to be conjugate fields. Prove that \(K\) and \(L\) are conjugate if and only if \(G(E/K)\) and \(G(E/L)\) are conjugate subgroups of \(G(E/F)\text{.}\)
Let \(F\) be a field such that \(\chr(F) \neq 2\text{.}\) Prove that the splitting field of \(f(x) = a x^2 + b x + c\) is \(F( \sqrt{\alpha}\, )\text{,}\) where \(\alpha = b^2 - 4ac\text{.}\)
Let \(K\) be the splitting field of a polynomial over \(F\text{.}\) If \(E\) is a field extension of \(F\) contained in \(K\) and \([E:F] = 2\text{,}\) then \(E\) is the splitting field of some polynomial in \(F[x]\text{.}\)
is irreducible over \({\mathbb Q}\) for every prime \(p\text{.}\) Let \(\omega\) be a zero of \(\Phi_p(x)\text{,}\) and consider the field \({\mathbb Q}(\omega)\text{.}\)
Show that \(\omega, \omega^2, \ldots, \omega^{p-1}\) are distinct zeros of \(\Phi_p(x)\text{,}\) and conclude that they are all the zeros of \(\Phi_p(x)\text{.}\)
Clearly \(\omega, \omega^2, \ldots, \omega^{p - 1}\) are distinct since \(\omega \neq 1\) or 0. To show that \(\omega^i\) is a zero of \(\Phi_p\text{,}\) calculate \(\Phi_p( \omega^i)\text{.}\)
The conjugates of \(\omega\) are \(\omega, \omega^2, \ldots, \omega^{p - 1}\text{.}\) Define a map \(\phi_i: {\mathbb Q}(\omega) \rightarrow {\mathbb Q}(\omega^i)\) by
where \(a_i \in {\mathbb Q}\text{.}\) Prove that \(\phi_i\) is an isomorphism of fields. Show that \(\phi_2\) generates \(G({\mathbb Q}(\omega)/{\mathbb Q})\text{.}\)
Show that \(\{ \omega, \omega^2, \ldots, \omega^{p - 1} \}\) is a basis for \({\mathbb Q}( \omega )\) over \({\mathbb Q}\text{,}\) and consider which linear combinations of \(\omega, \omega^2, \ldots, \omega^{p - 1}\) are left fixed by all elements of \(G( {\mathbb Q}( \omega ) / {\mathbb Q})\text{.}\)
Let \(F\) be a finite field or a field of characteristic zero. Let \(E\) be a finite normal extension of \(F\) with Galois group \(G(E/F)\text{.}\) Prove that \(F \subset K \subset L \subset E\) if and only if \(\{ \identity \} \subset G(E/L) \subset G(E/K) \subset G(E/F)\text{.}\)
Let \(F\) be a field of characteristic zero and let \(f(x) \in F[x]\) be a separable polynomial of degree \(n\text{.}\) If \(E\) is the splitting field of \(f(x)\text{,}\) let \(\alpha_1, \ldots, \alpha_n\) be the roots of \(f(x)\) in \(E\text{.}\) Let \(\Delta = \prod_{i \lt j} (\alpha_i - \alpha_j)\text{.}\) We define the discriminant of \(f(x)\) to be \(\Delta^2\text{.}\)
If \(f(x) = x^2 + b x + c\text{,}\) show that \(\Delta^2 = b^2 - 4c\text{.}\)
