proof that the cyclotomic polynomial is irreducible
We first prove that . The field extension of is the splitting field of the polynomial , since it splits this polynomial and is generated as an algebra by a single root of the polynomial. Since splitting fields are normal, the extension is a Galois extension. Any element of the Galois group, being a field automorphism, must map to another root of unity of exact order . Therefore, since the Galois group of permutes the roots of , it must fix the coefficients of , so by Galois theory these coefficients are in . Moreover, since the coefficients are algebraic integers, they must be in as well.
Let be the minimal polynomial of in . Then has integer coefficients as well, since is an algebraic integer. We will prove by showing that every root of is a root of . We do so via the following claim:
Claim: For any prime not dividing , and any primitive root of unity , if then .
This claim does the job, since we know , and any other primitive root of unity can be obtained from by successively raising by prime powers not dividing a finite number of times11Actually, if one applies Dirichlet’s theorem on primes in arithmetic progressions here, it turns out that one prime is enough, but we do not need such a sharp result here..
To prove this claim, consider the factorization for some polynomial . Writing for the ring of integers of , we treat the factorization as taking place in and proceed to mod out both sides of the factorization by any prime ideal of lying over . Note that the polynomial has no repeated roots mod , since its derivative is relatively prime to mod . Therefore, if , then , and applying the power Frobenius map to both sides yields . This means that cannot be 0 in , because it doesn’t even equal . However, is a root of , so if it is not a root of , it must be a root of , and so we have , as desired.
|Title||proof that the cyclotomic polynomial is irreducible|
|Date of creation||2013-03-22 12:38:04|
|Last modified on||2013-03-22 12:38:04|
|Last modified by||djao (24)|