Exercise 5.1.11

Question

\def\imp{\Rightarrow}
\def\gcro{\mbox{[\hspace{-.15em}[}}% intervalles d'entiers 
\def\dcro{\mbox{]\hspace{-.15em}]}}

ewcommand{\be} {\begin{enumerate}}

ewcommand{\ee} {\end{enumerate}}

ewcommand{\deb}{\begin{eqnarray*}}

ewcommand{\fin}{\end{eqnarray*}}

ewcommand{\ssi} {si et seulement si }

ewcommand{\D}{\mathrm{d}}

ewcommand{\Q}{\mathbb{Q}}

ewcommand{\Z}{\mathbb{Z}}

ewcommand{\N}{\mathbb{N}}

ewcommand{\R}{\mathbb{R}}

ewcommand{\C}{\mathbb{C}}

ewcommand{\F}{\mathbb{F}}

ewcommand{\U}{\mathbb{U}}

ewcommand{e}{\,\mathrm{Re}\,}

ewcommand{\im}{\,\mathrm{Im}\,}

ewcommand{\ord}{\mathrm{ord}}

ewcommand{\Gal}{\mathrm{Gal}}

ewcommand{\legendre}[2]{\genfrac{(}{)}{}{}{#1}{#2}}

Suppose that $f \in F[x]$ is irreducible of degree $n>0$, and let $L$ be the splitting field of $f$ over $F$.
\begin{enumerate}
\item[(a)] Prove that $n \mid [L:F]$.
\item[(b)] Give an example to show that $n = [L:F]$ can occur in part (a).
\end{enumerate}

Richard Ganaye · Accepted Answer

\def\imp{\Rightarrow}
\def\gcro{\mbox{[\hspace{-.15em}[}}% intervalles d'entiers 
\def\dcro{\mbox{]\hspace{-.15em}]}}

ewcommand{\be} {\begin{enumerate}}

ewcommand{\ee} {\end{enumerate}}

ewcommand{\deb}{\begin{eqnarray*}}

ewcommand{\fin}{\end{eqnarray*}}

ewcommand{\ssi} {si et seulement si }

ewcommand{\D}{\mathrm{d}}

ewcommand{\Q}{\mathbb{Q}}

ewcommand{\Z}{\mathbb{Z}}

ewcommand{\N}{\mathbb{N}}

ewcommand{\R}{\mathbb{R}}

ewcommand{\C}{\mathbb{C}}

ewcommand{\F}{\mathbb{F}}

ewcommand{\U}{\mathbb{U}}

ewcommand{e}{\,\mathrm{Re}\,}

ewcommand{\im}{\,\mathrm{Im}\,}

ewcommand{\ord}{\mathrm{ord}}

ewcommand{\Gal}{\mathrm{Gal}}

ewcommand{\legendre}[2]{\genfrac{(}{)}{}{}{#1}{#2}}

\begin{proof}
\begin{enumerate}
\item[(a)]
Let $\alpha \in L$ a root of $f$. Then $F \subset F(\alpha) \subset L$, thus
$$[L:F] = [L:F(\alpha)]\ [F(\alpha) : F].$$ 
As $f$ is the minimal polynomial of $\alpha$, $[F(\alpha):F] = \deg(f) = n$, thus
$n \mid [L:F]$.

\item[(b)]
In Exercise 6, we have seen that $f = x^4-4x^2+2$, of degree $n=4$,  has for splitting field $L  = \Q(\sqrt{2}+ \sqrt{3}) = \Q(\sqrt{2},\sqrt{3})$, of degree 4 over $\Q$.
Here $[L : \Q] = 4 = \deg(f)$, the equality in relation (a) is so a possibility.
\end{enumerate}
\end{proof}

Exercise 5.1.11

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Exercise 5.1.11

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