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3.5. Roots of Unity 103

9. The nth cyclotomic polynomials Q,,(t) is the product of all the linear
   polynomials (t - (‘) where C ranges over the primitive nth roots of
   unity. Show that
   Pn(t) = jlIQd(t)

where the product is taken over all the divisors d of n.

(It turns out that the coefficients of Qn(t) are integers and that these

polynomials are irreducible over Q.)

10. Verify that

(a) &2(t) = t + 1

(b) &4(t) = t2 + 1

(c) Qs(t) = t2 -t + 1

(4 &s(t) = t4 + 1

(e) Qp(t) = tp-’ + tpv2 +... + t + 1 when p is prime.

11. Compute the cyclotomic polynomials Qs, Qib, Q12, Qi4, Qis and
    &IS.


12. (a) If k is even, show that C&k(t) = Qk(t2).
    (b) If k is odd and exceeds 2, show that &2k(t) = Qk(-t).


13. Find the nth roots of -1 and factor the polynomial t” + 1 over C, R
    and Q.


14. Let c = r(cos0 + isin 8) be a complex number. Find the nth roots of
    c and factor the polynomial t” - c over C. In particular, find the nth
    roots of 2 and factor t” - 2 over C, R and Q.


15. Show that, if k 1 n, then Pk(t) I P,(t) over C, R and Q. Test the
    conjecture: if k and m are divisors of n, then Pk(t)Pm(t) I P,(t).


16. Let C be a primitive 5th root of unity. Show that u = c2 + C3 and
    v = (‘l + C4 are zeros of the polynomial t2 + t - 1.


17. Let (’ be a primitive 7th root of unity. Show that u = C3 + C5 + C6
    and v = C + C2 + C4 are zeros of the quadratic t2 + t,+ 2.


18. Let p be any prime. We say that a is a quadratic residue modulo p
    if there is some number x such that x2 E a (mod p). Otherwise, we
    call a a quadratic nonresidue.

(a) Verify that the quadratic residues modulo 5 are 1 and 4 and

those modulo 7 are 1, 2, 4.

(b) Find the quadratic residues modulo 11, 13, 17.