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Algebra

Cool goIITian

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9 Feb 2010 22:25:51 IST

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prove that the pth power of the nth root of unity always vanishes until its a factor of n ????

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prove that the pth power of the nth root of unity always vanishes until its a factor of n ????

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Anant Kumar

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Joined: 10 Jul 2008

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9 Feb 2010 23:08:00 IST

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An $n$ -th root of unity is of the form $e^{i\frac{2k\pi}{n}}$ where $k=0,1, \ldots, (n-1)$ .
So its $p$ -th power is $e^{i\frac{2pk\pi}{n}}$ which cannot vanish for any $p$ .
Geometrically, you can see that a root of unity lies on the unit circle. Raising it to any power rotates it by certain angle leaving the length unchanged. So no matter how many times you rotate it, it cannot end up at the origin.

(that means you should check the question.)

ghfh bgd

Cool goIITian

Joined: 21 Apr 2009

Posts: 70

4 Mar 2010 02:11:40 IST

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i was also also struck here but its a question from fitjee .

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