4 февр. 2016 г. · Sometimes, the n-th cyclotomic polynomial is defined as the minimal polynomial of ζn over Q. Then Φn divides Xn−1 in Q[X], since it is a minimal ...

31 окт. 2013 г. · In the first case we take k=(q+n)/2 which is easily seen to be in this domain. In the second case take k=q/2, also clearly in this domain.

3 нояб. 2019 г. · Suppose g(ζp)=0. Then ζ is a root of g(xp) and so f(x) must divide g(xp) in Z[x]. So, g(xp)=f(x)h(x) for some h(x)∈Z[x]. Reducing mod p gives ˜g ...

1 нояб. 2015 г. · Hence Φ8(x)=x4+1 is irreducible over Q. The other factorisation you gave is not over Q; this would rather work with x4+4=(x2−2x+2)(x2+2x+2).

6 июл. 2011 г. · Let Φn(x) be the usual cyclotomic polynomial (minimal polynomial over the rationals for a primitive nth root of unity). There are many well- ...

24 июн. 2024 г. · definition of cyclotomic polynomials ... In every reference I have encountered, the cyclotomic polynomials are defined in terms of roots of unity.

18 дек. 2011 г. · A polynomial is solvable iff its splitting field is solvable iff its Galois group is solvable. The Galois group of the cyclotomic polynomial Φn( ...

21 окт. 2013 г. · One can show that the cyclotomic polynomial Φn(X) is irreducible over Fp precisely when p has multiplicative order φ(n) modulo n.

15 дек. 2023 г. · I can prove that the primitive nth roots of unity are roots of the nth cyclotomic polynomial. Proof: All primitive nth roots of unity ζn share ...

30 апр. 2018 г. · The nth cyclotomic polynomial remains irreducible when reduced modulo p if and only if p is a generator of Z×n. Suppose that is not the case, ...