Math 6320, Assignment 4
Due in class: Thursday, March 24
1. Let p be an odd prime, and set ζ = e2πi/p . Prove:
p−1
(a)
p−1
∏ (ζ i − 1) = ∏ (1 − ζ i ) = p
i=1
i=1
p−1
(b)
∏ (ζ i − ζ j ) = p p
i, j=0
i6= j
p−1
(c)
∏ (ζ i − ζ j )2 = (−1)(p−1)/2 p p−2
i, j=1
i< j
2. For p and ζ as above, prove that the discriminant of Φ p (x) is a square in Q(ζ ).
3. Let p be a prime. Prove:
√
(a) 2 ∈ Q(e2πi/8 )
√
(b) p ∈ Q(e2πi/p ) if p ≡ 1 mod 4
√
(c) p ∈ Q(e2πi/4p ) if p ≡ 3 mod 4
4. Let p be a prime integer, and n a positive integer with p - n.
(a) Let a be an integer. Prove that p divides Φn (a) if and only if a has order n in (Z/p)× .
(b) Prove that p divides Φn (a) for some integer a if and only if p ≡ 1 mod n.
5. Fix a positive integer n. Prove that the set
{p : p is prime and p divides Φn (a) for some a}
is infinite. Conclude that the arithmetic sequence 1, 1 + n, 1 + 2n, . . . contains infinitely many primes.
6. Let G be the Galois group of x5 − 7 ∈ Q[x]. Is G solvable? Is it abelian?
7. If a and b are rational numbers satisfying a2 + b2 = 1, use Hilbert’s Theorem 90 to prove that
a=
s2 − t 2
s2 + t 2
and
b=
2st
s2 + t 2
for some s,t ∈ Q .
This shows that any right triangle with integer sides has sides of length
d(s2 − t 2 ),
2dst,
d(s2 + t 2 ),
for d, s,t ∈ Z .
8. Prove that the product of two nonsquares in the multiplicative group F×
p is a square.
(This implies, for example, that at least one of 2, 3, 6 is a square in F×
p for p > 5.)
√
√
√
9. (a) Prove that f (x) = x4 − 10x2 + 1 factors over each of Z[ 2], Z[ 3], and Z[ 6].
(b) Show that f (x) is irreducible in Q[x].
(c) Use the previous problem to conclude that f (x) factors in F p [x] for each prime p.
10. Let σ and τ be automorphisms of the field C(x) that fix C and satisfy
σ (x) = e2πi/3 x,
τ(x) = 1/x.
Prove that hσ , τi is isomorphic to the symmetric group S3 , and determine the fixed field.