Talk:Cyclotomic field

Properties
That is not always a normal basis. Any linear dependence among them would kill their ability to form a normal basis. Linear dependences of the form a+b=0 occur whenever n is a multiple of 4, since then 1 + n/2 is relatively prime to n and the 1 + n/2 power of any primitive nth root of unity is just its additive inverse. (This example can be generalized to n whenever n is not squarefree; then one gets products of a primitive nth root of unity with all pth roots of unity adding up to 0, whenever p is prime and p^2 | n.)

DavidLHarden (talk) 05:39, 17 April 2008 (UTC)

Fermat's last theorem
You'll want to add that the factorization you gave of x^n + y^n is only valid for when n is a prime number

Actually the factorization as given is valid iff n is odd. RJChapman 15:11, 14 August 2007 (UTC)


 * By chance did something related and then asked myself the same question, for an even n we have x^n - y^n = the product. See also here: https://math.stackexchange.com/q/2334063/4414 Jan Burse (talk) 21:19, 23 June 2017 (UTC)


 * The question is answered there and is not particularly relevant to this article. 67.198.37.16 (talk) 18:40, 6 October 2020 (UTC)

Gaussian rational
The article Gaussian rational states nothing unique for Q(ζ4) among others Q(ζn). Any cyclotomic field is "neither ordered nor topologically complete", is "an example of an algebraic number field" and "a Galois extension of the field of rational numbers". Q(ζ4) is a quadratic field, but Q(ζ3) also is Q(√−3). Surely, its ring of integers has an independent notability, but Q(ζ3) has a notable ring of integers too. Has that article to be merged here to avoid virtually a duplicate description? Incnis Mrsi (talk) 19:38, 24 January 2012 (UTC)


 * As of 2015, a section on Ford Spheres was added. This seems not to be a property held in common with other cyclotomic fields, and thus, the merge suggestion evaporates. 67.198.37.16 (talk) 18:35, 6 October 2020 (UTC)