Talk:Root of unity
|WikiProject Mathematics||(Rated B-class, Mid-importance)|
if n is not divisible by the characteristic of the field
My wording about exactly n nth roots was a bit reckless, but I did not claim that zn = 1 always has distinct roots. BTW I just learned about a (weaker) definition of cyclotomically closed field, which apparently is exactly what we need here. Incnis Mrsi (talk) 18:03, 5 April 2013 (UTC)
It would be nice to have a "Geometric interpretation" section that describes the fact that all n roots of unity lie at equidistant points on the unit circle (centered at the origin) in the cartesian/complex plane, i.e., they lie on the vertices of a regular n-gon inscribed within the unit circle. But I don't know where the most appropriate place is to add such a section in the article. — Loadmaster (talk) 19:00, 30 September 2013 (UTC)
- Well, multiplying any combination of the nth roots of 1 always produces one of the (other) roots (or 1) on the complex unit circle. Does that qualify as "special"? There is no deeper meaning, since these are just some of the properties of the unit multiplier (1) in the complex field. — Loadmaster (talk) 19:15, 19 August 2014 (UTC)
Random observation while I was passing through: should the opening paragraph specify that n is positive? After all, raising any complex number (save 0) to the 0th power yields 1. 0 is obviously an integer, but that doesn't mean that the number is a root of unity. (The definition below does a better job, specifying that n must be positive.)
I hate do not understanding stuff!
In this case, what I don't understand is this assertion in the article:
- Generally, z ∈ R can be considered for any field R
I'm sure this is a stupid question*, but does this mean that in general, a complex number is a real? I guess it depends on the definition of "considered."
Even if not, why talk about complex numbers that are points on the real number line? The imaginary part is zero, so it's a degenerative, trivial complex number.
For someone who hates not understanding stuff, I sure do it a lot.
- I can't stand when people preface questions with that when asking me, but now I see why they do. --Verdana♥Bold 07:57, 4 March 2017 (UTC)
- You are confusing R, a variable, which, here, denotes an arbitrary ring, with the standard symbol for the field of real numbers, which is or R. Nevertheless, I agree that the formulation was confusing, and not only because of the use of R (which was unnecessary). I have thus rewritten the paragraph. D.Lazard (talk) 08:57, 4 March 2017 (UTC)