% Feuille de TD 10 Algebre et Arithmetique, Licence 3, 2008-2009

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\title{$\boxed{\textsc{FEUILLE DE TRAVAUX DIRIG\'ES 10}}$}
\author{\bf Irr\'eductibilit\'e des polyn\^omes }



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{\sc Alg\`ebre et Arithm\'etique} \hfill {\sc Math\'ematiques L3}

{Universit\'e de Nice Sophia-Antipolis} \hfill {Ann\'ee 2008-2009}\\

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\begin{ex}[Polyn\^omes irr\'eductibles]$\ $

Parmi les polyn\^omes suivants, lesquel sont irr\'eductibles dans $\QQ[X]$ ? 
$$X^3+14, \ X^2+49,\ X^2-49, \ X^3+4$$
\end{ex}


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\begin{ex}[Division dans une extension]$\ $

Soit $K\subset L$ une extension de corps et $P,\, Q \in K[X]$. Montrer l'\'equivalence 
$$P\ \textrm{divise} \ Q\ \textrm{dans}\   K[X] \ \Longleftrightarrow \ P \ \textrm{divise}\ Q \ \textrm{dans}\ L[X].$$

\end{ex}
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\begin{ex}[Extensions et nombres premiers]$\ $

Montrer que si $p$ et $q$ sont des entiers premiers, alors $\QQ[\sqrt{p}+\sqrt{q}]=\QQ[\sqrt{p},\sqrt{q}]$. 
\end{ex}
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\begin{ex}[Factorisation]$ \ $ 


Factoriser sur $\QQ$, $\RR$ et $\CC$ le polyn\^ome $X^4+4$. 


\end{ex}

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\begin{ex}[Polyn\^omes minimaux]$\ $


Donner les polyn\^omes minimaux sur $\QQ$ de $\sqrt{1+\sqrt{3}}$ et de  $\sqrt{4+2\sqrt{3}}$.
\end{ex}
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\begin{ex}[Polyn\^omes irr\'eductibles II]$\ $

Le polyn\^ome $X^4+X^3+2X+1$ est-il irr\'eductible dans $\ZZ[X]$ ?

Le polyn\^ome $X^4-10X^3+21X^2-10X+11$ est-il irr\'eductible dans $\ZZ[X]$ ?
\end{ex}

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\begin{ex}[Polyn\^omes irr\'eductibles II]$\ $

Soit $p$ un nombre premier, montrer que le polyn\^ome $X^{p-1}+X^{p-2}+\cdots+X+1$ est irr\'eductible dans $\ZZ[X]$ ?

\underline{Indication :} Utiliser $X=Y+1$. 
\end{ex}



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