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10. SOLVABILITY

10.4. Cyclotomic Polynomials

In this section, we will study factors of the polynomial xn

1. Recall that a primitive nth root of unity in a field K is an element  2 K such that n D 1 but d ¤ 1 for d < n. The primitive nth roots of 1 in C are the numbers e2ir=n, where r is relatively prime to n. Thus the number of primitive nth roots is '.n/, where ' denotes the the Euler function.

Definition 10.4.1. The nth cyclotomic polynomial  n.x/ is defined by

Y n.x/ D f.x

/ W  a primitive nth root of 1 in Cg:

A priori,  n.x/ is a polynomial in CŒx, but, in fact, it turns out to have integer coefficients. It is evident that  n.x/ is a monic polynomial of degree '.n/. Note the factorization:

Y

xn

1 D

f.x

/ W  an nth root of 1 in Cg

Y

Y

D

f.x

/ W  a primitive d th root of 1 in Cg d divides n

Y

D d .x/:

(10.4.1)

d divides n

Using this, we can compute the polynomials  n.x/ recursively, begin ning with  1.x/ D x 1. See Exercise 10.4.2.

Proposition 10.4.2. For all n, the cyclotomic polynomial  n.x/ is a monic polynomial of degree '.n/ with integer coefficients.

Proof. The assertion is valid for n D 1 by inspection. We proceed by induction on n. Fix n > 1, and put

Y

f .x/ D d .x/;

d
n

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10.4. CYCLOTOMIC POLYNOMIALS

477

Note that the splitting field of xn 1 coincides with the splitting field of  n.x/ and is equal to Q./, where  is any primitive nth root of unity (Exercise 10.4.3). Next, we wish to show that  n.x/ is irreducible over Q, so Q./ is a field extension of degree '.n/ over Q. First we note the following:

Lemma 10.4.3. The following statements are equivalent: (a) n.x/ is irreducible.

(b) If  is a primitive nth root of unity in C, f is the minimal polynomial of  over Q, and p is a prime not dividing n, then p is a root of f .

(c) If  is a primitive nth root of unity in C, f is the minimal polynomial of  over Q, and r is relatively prime to n, then r is a root of f .

(d) If  is a primitive nth root of unity in C, and r is relatively prime to n, then  7! r determines an automorphism of Q./ over Q.

Proof. Exercise 10.4.5.

n

Theorem 10.4.4.  n.x/ is irreducible over Q.

Proof. Suppose that  n.x/ D f .x/g.x/, where f; g 2 ZŒx, and f is irreducible. Let  be a root of f in C; thus  is a primitive nth root of unity and f is the minimal polynomial of  over Q. Suppose that p is a prime not dividing n. According to the previous lemma, it suffices to show that p is a root of f .

Suppose that p is not a root of f . Then, necessarily, p is a root of g, or, equivalently,  is a root of g.xp/. It follows that f .x/ divides g.xp/, because f is the minimal polynomial of . Reducing all polynomials modulo p, we have that Q f .x/ divides Qg.xp/ D . Qg.x//p. In particular,

Q f .x/ and Qg.x/ are not relatively prime in ZpŒx. Now, it follows from

Q n.x/ D Q f .x/ Qg.x/ that Q n.x/ has a multiple root, and, therefore, xn

1

has a multiple root over Zp. But this cannot be so, because the derivative of xn 1 in ZpŒx, namely Œnxn 1, is not identically zero, since p does not divide n, and has no roots in common with xn 1. This contradiction shows that, in fact, p is a root of f .

n

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10. SOLVABILITY

Corollary 10.4.5. The splitting field of xn 1 over Q is Q./, where  is a primitive nth root of unity. The dimension of Q./ over Q is '.n/, and the Galois group of Q./ over Q is isomorphic to ˚.n/, the multiplicative group of units in Zn.

Proof. The irreducibility of  n.x/ over Q implies that dimQ.Q.// D '.n/. We can define an injective homomorphism of G D AutQ.Q./ into ˚.n/ by  7! Œr if ./ D r . This map is surjective since both groups have the same cardinality.

n

Proposition 10.4.6. Let K be any field whose characteristic does not divide n. The the Galois group of xn 1 over K is isomorphic to a subgroup of ˚.n/. In particular, if n is prime, then the Galois group is cyclic of order dividing n

1.

Proof. Exercise 10.4.6.

n

Exercises 10.4 10.4.1. If  is any primitive nth root of unity in C, show that the set of all primitive nth roots of unity in C is fr W r is relatively prime to ng.

10.4.2. Show how to use Equation 10.4.1 to compute the cyclotomic polynomials recursively. Compute the first several cyclotomic polynomials by this method.

10.4.3. Note that the splitting field of xn 1 coincides with the splitting field of  n.x/ and is equal to Q./, where  is any primitive nth root of unity.

10.4.4. Show that Q.x/ \ CŒx D QŒx.

10.4.5. Prove Lemma 10.4.3.

10.4.6. Prove Proposition 10.4.6. Hint: Let E be a splitting field of xn

1

over K. Show that the hypothesis on characteristics implies that E contains a primitive nth root of unity . Define an injective homomorphism from G D AutK.E/ into ˚.n/ by  7! Œr, where ./ D r . Show that this is a well–defined, injective homomorphism of groups. (In particular, check the proof of Corollary 10.4.5.)

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