Bonjour,
Qₙ (X) = X²ⁿ - 2cos(nα)Xⁿ + 1
Factorisation sur C[X]
Qₙ(X) = =(Xⁿ − exp(inα))(Xn − exp(−inα))
or : Xⁿ − exp(inα) = exp(inα)(([tex]\frac{X}{exp(ia)}[/tex] )ⁿ - 1)
= exp(inα)[tex]\displaystyle \prod _{k=0}^{n-1}(\frac{X}{exp(ia)} -exp(\frac{2ik\pi }{n} ))[/tex]
[tex]=\displaystyle \prod _{k=0}^{n-1}(X - exp(i(a+\frac{2k\pi }{n} )))[/tex]
De la même manière on a :
[tex](X^{n} - exp(ina)) = \displaystyle\prod _{k=0}^{n-1}(X - exp(i(-a + \frac{2k\pi }{n} )))[/tex]
Ainsi on obtient :
[tex]Q_{n}(X) =\displaystyle\prod _{k=0}^{n-1}(X - exp(i(a+\frac{2k\pi }{n} )))\times\displaystyle \prod _{k=0}^{n-1}(X-exp(i(-a+\frac{2k\pi }{n})))[/tex]
[tex]Q_{n}(X)= \displaystyle\prod _{k=0}^{n-1}(X^{2} -2cos(a +\frac{2k\pi }{n} )X+1)[/tex]
