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Grey Council: I dunno why, but Archman is doing the HSC this year...
wtf, how do we even stand a chance against him.
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Don't worry there's at least 6 mths to go, to build up your strength to tackle question 8 types. Invariably these questions rely on a series of quite basic steps strung together.
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If w, w^2, ...,w^n are the n complex roots of 1, with w being of smallest positive argument, what is the condition that need to be satisfied for W=w^k, so that W, W^2, W^3,...,W^n are also the n roots of unity; and why?
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Here's why gcd(k,n)=1
1) By the factor theorem, you can't have more than n roots of z^n=1.
2) Easy to show that each W^j is also a root.
3) Must prove that for n>=j1>j2>=1 W^j1 not.=W^j2
Assume false, ie. W^j1 =W^j2
then cis(j1*k2pi/n)=cis(j2*k2pi/n)
hence j1*k2pi/n - j2*k2pi/n = m2pi where m is an integer
therefore k(j1-j2) = mn
thus n divides k(j1-j2)
but gcd(k,n)=1 hence n divides (j1-j2) * contradiction*.
Therefore W, W^2, W^3,...,W^n are the same n roots of unity.
Now this should end the nth root of unity story. I say this tongue in cheek with apologies to Francis Fukuyama's seminal work End of History when the Berlin Wall was torn down: history doesn't end, it moves on. No doubt the examiners will tell and retell the story starting with the phrase "Question 8"
Really tried to stir the discussion to above, but unfortunately it became a brawl.
