Submitted by mrmonkey on Mon, 04/09/2018 - 21:20
For the STEP 3 complex numbers assignment q1 part 2:
$x^4+1=0\\
x^4=-1\\
x=e^{i\frac{\pi}{4}},e^{i\frac{3\pi}{4}},e^{-i\frac{\pi}{4}},e^{-i\frac{3\pi}{4}}\\$
We know all the solutions satisfy $x^n=1$, so taking one of them:
$(e^{i\frac{\pi}{4}})^n=1\\
e^{i\frac{n\pi}{4}}=1\\
\frac{n\pi}{4}=2k\pi\\
n=8k\\$
For k being an integer. As they are all primitive roots, we want smallest k where n is a positive integer, so taking $k=1$, we get $n=8$.
If you use the $e^{i\frac{3\pi}{4}}$ solution instead we get:
$n=\frac{8}{3}k$, and as we want the smallest value of k for n to be an integer, $k=3$, to obtain $n=8$
If this method is wrong, at which point is it wrong?
thanks
PS on the solutions, the blue part should read $...x^4=-1$ I believe?
Alternative method
This is essentially the same as method I was getting at in the blue "alternatively" part (and yes, it should say $x^4=-1$, thanks!!
To complete the method, you should then write down the 4 primitive roots of $x^8=1$ (write down is ok, you don't need to show that they are primitive) and then show that $C_8(x)$ is actually $x^4+1$.