Sample HTML file: produced with lwarp, with mathematical formulae displayed with MathJax, and xymatrix commutative diagrams displayed with XyJax-v3
6.2 The Galois correspondence for \(f(t)=t^3-2\), over the rational field
Let \(f(t)=t^3-2 \in \Q [t]\). Let \(\w =e^{\frac {2\pi i}{3}}\). Let \(\a =\sqrt [3]{2}\), \(\beta =\a \w \) and \(\gamma =\a \w ^2\). Hence the set of roots of \(f\) is \(\{\a ,\b ,\g \}\). The splitting field of \(f\), over \(\Q \) is:
\[\Q (\a ,\b ,\g )=\Q (\a ,\w ).\]
The monomorphism of groups,
\(\seteqnumber{0}{}{3}\)\begin{align*} \theta \colon \Gamma (f,\Q ) &\longrightarrow \Sym (\{\a ,\b ,\g \})\\ \tau & \longmapsto \left ( \begin{CD} \{\a ,\b ,\g \} &\longrightarrow &\{\a ,\b ,\g \}\\ a &\longmapsto &\tau (a) \end {CD}\right ), \end{align*} is, in this case, an isomorphism. The diagram of subgroups of \(\Gamma (f,\Q ) \cong \Sym (\{\a ,\b ,\g \})\) is below (note that inclusions go in the direction of arrows):
\[\xymatrix { && \Sym (\{\a ,\b ,\gamma \} )\\ & \{\id , (\a \b \g ), (\a \g \b )\} \ar [ur] \\ & & \{\id ,(\a \b )\}\ar [uu] &\{\id ,(\a \g )\}\ar [uul] &\{\id , (\b \g )\}\ar [uull]\\ && \{\id _R\}\ar [uul]\ar [u] \ar [ur] \ar [urr] } \]
The corresponding diagram of intermediate fields \(\Q \subseteq L \subseteq \Q (\a ,\b ,\g )\) is:
\[\xymatrix { && \Q \\ & \Q (\w ) \ar @{<-}[ur] \\ & & \Q (\g )\ar @{<-}[uu] & \Q (\b )\ar @{<-}[uul] & \Q (\a )\ar @{<-}[uull]\\ &&\Q (\a ,\b ,\g )\ar @{<-}[uul]\ar @{<-}[u] \ar @{<-}[ur] \ar @{<-}[urr] } \]
Also note that \(\{\id _R\}\), \(\Sym (R)\) and \({\{\id , (\a \b \g ), (\a \g \b )\}}\cong A_3\) are the only normal subgroups of \(\Sym (R)\). So, respectively,
\(\seteqnumber{0}{}{3}\)\begin{equation} \label {normalext} \Q (\a ,\b ,\g ):\Q ,\qquad \Q :\Q \quad \textrm { and } \quad \Q (\w ):\Q \end{equation}
are the only normal extensions of \(\Q \) contained in \(\Q (\a ,\b ,\g )\). Note that the fact that the extensions in (4) are are normal also follows from the fact that they are the splitting fields of \(p(t)=t^3-2\), \(q(t)=t\) and \(r(t)=t^2+t+1\) (which has \(\w \) and \(\w ^2\) as roots), over \(\Q \).
-
Remark 3. Also note that we have a series of subfields of \(\C \) (we use \(\leq \) to denote subfield):
\[\Q \leq \Q (\w ) \leq \Q (\a ,\b \,\g )=\Q (\a ,\w ).\]
Note that:
-
• \(\Q (\w ): \Q \) is a normal extension (since it is the splitting field of \(t^2+t+1\), over \(\Q \)).
-
• \(\Q (\w ,\a ):\Q (\w )\) is also a normal extension. This is because \(\Q (\w ,\a )\) is the splitting field of \(t^3-2\) over \(\Q (\w )\).
And then it follows that:
-
• \(\Gamma \big (\Q (\w ,\a ),\Q (\w )\big )\) is normal in \(\Gamma (\Q (\w ,\a ): \Q )\),
-
• we have a series of subgroups of \(\Gamma \big ( \Q (\w ,\a ): \Q \big )\):
\[\{e\}=\Gamma \big ( \Q (\w ,a): \Q (\w ,\a ) \big )\trianglelefteq \Gamma \big ( \Q (\w ,\a ): \Q (\w ) \big ) \trianglelefteq \Gamma \big ( \Q (\w ,\a ): \Q \big ),\]
-
• The quotient groups can be explicitly determined:
-
– \(\Gamma \big ( \Q (\w ,\a ): \Q \big ) / \Gamma \big ( \Q (\w ,a): \Q (\w ) \big ) \cong \Gamma \big (\Q (\w ): \Q )\cong \Z _2\).
Where the last equation follows since \(\Q (\w ): \Q \) is a normal extension of degree 2, since it is the splitting field of the irreducible polynomial \(t^2+t+1\), over \(\Q \). This \(\Z _2\) is an abelian group.
-
– \(\Gamma \big ( \Q (\w ,\a ): \Q (\w ) \big ) / \Gamma \big ( \Q (\w ,\a ): \Q (\w ,\a ) \big ) \cong \Gamma \big (\Q (\w ,\a ): \Q (\w ))\cong \Z _3. \)
Where the last equation follows since \(\Q (\w ,\a ): \Q (\w )\) is a normal extension of degree 3. This is because it is the splitting field of the polynomial \(t^3-2\), which is irreducible over \(\Q (\w )\). This is an abelian group.
What was just shown is a general patern that exists any time we compute the splitting field of a polynomial that is soluble by radicals.
-
-