Exercise 11.1.2

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 $f,g \in F[x]$ are polynomials, not both zero, and let $h$ be their greatest common divisor as computed in $F[x]$. Now let $L$ be an extension field of $F$. Prove that $h$ is the greatest common divisor of $f,g$ when considered as polynomial in $L[x]$.

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}
Recall the definition of the gcd in $F[x]$, where $f,g,h \in F[x]$ :

$h = f\wedge g$ if and only if

(i) $h$ is monic (or zero).

(ii) $h \mid f,\  h \mid g$

(iii) $\forall p \in F[x], (p \mid f,\  p \mid g) \Rightarrow p \mid h$.

Let $L$ be an extension field of $F$. $f$ satisfies (i)(ii) in $L[x]$, since $h$ divides $f$ in $F[x]$ if and only if $h$ divides $f$ in $L[x]$.

\be 
\item[$\bullet$] If
$$\forall p \in L[x], (p \mid f,\  p \mid g) \Rightarrow p \mid h,$$
since $F[x] \subset L[x]$,
$$\forall p \in F[x], (p \mid f,\  p \mid g) \Rightarrow p \mid h.$$
So, $h = \gcd(f,g)$ in $L[x]$ implies that $h = \gcd(f,g)$ in $F[x]$.

\item[$\bullet$] Conversely, suppose that $h = \gcd(f,g)$ in $F[x]$.

By B\'ezout's Theorem, there exists $u,v \in F[x]$ such that $h = uf +vg$. Consequently, if $q \in L[x]$ divides $f$ and $g$ in $L[x]$, $q$ divides $h$ in $L[x]$. So $h$ satisfies (iii), therefore $h$ is the greatest common divisor of $f,g$ in $L[x]$.
\ee
\end{proof}

Exercise 11.1.2

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Exercise 11.1.2

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