What is meaning of 2n factors?

Question

Can any one explain what does 2n factors mean and how it's been written n factors in next step. I tried to figure out but couldn't make out.

Answer 1

It means that there are 2n terms, each term being a factor of the form $(1+\omega^2)$. In the second step, they've grouped two terms together, for eg. $[(1+\omega)(1+\omega^2)]$ is one term and so, there are n such factors.

Answer 2

At the risk of making something more confusing than it should be.

$N = a_1*a_2*a_3* ...... *a_{2n}$ has $2n$ factors. $a_i = (1 + \omega^{2^{i - 1}})$

$N = b_1*b_2*b_3*......*b_n$ has $n$ factors. $b_i = a_{2i-1}*a_{2i} = (1+\omega^{2^{2i-2}})(1+ \omega^{2^{2i-1}})$

The next several lines rewrites $b_i$ into different forms.

Eventually it concludes $b_i = (1+\omega)(1 + \omega^2)$ for all $b_i$. So all the $b_i$ are equal.

The next line is

$N = b_1 * b_1* ......$ with $n$ factors and $b_1 = (1+\omega)(1+\omega^2)$

So $N = b_1^n = [ (1+\omega)(1+\omega^2)]^n$.

And that's the only reason we were counting the factors. So we'd know how many $[ (1+\omega)(1+\omega^2)]$ terms there were in the end.