Dr. Marques Sophie
Office 519
Algebra II
Fall Semester 2013
marques@cims.nyu.edu
Problem Set # 11
Due Wednesday december 4th in recitation
Exercise 1(?): 100 points
Define the n-th cyclotomic polynomial to be
Y
φn (x) =
(x − e
2iπk
n
)
1≤k≤n,gcd(k,n)=1
1. Prove that φn (x) =
n
Q x −1 .
d|n φd (x)
(Hint: Use a partition of {1, ..., n}.)
2. We have seen that φp is irreducible for p prime homework 3. We want to prove
that this is true in general.
(a) Prove that it is possible to write φn (x) = (a/b)f (x)g(x) which a and b
integers and f (x) and g(x) primitive polynomial in Z[x].
(b) What is the degree of φn (x)?
(c) Show that if we prove that if ξ is a root of f (x) then ξ a is also a root of
f (x) for any a ∈ (Z/nZ)× , then we are done. Prove then that it is enough
to prove that if ξ is a root of f (x) then ξ p is a root of f (x) for any prime
not dividing n.
(d) Suppose by contradiction that ξ p is not a root of f (x) for some p not dividing
n. Then prove that ξ is a root of g(xp ) and that d(x) = gcd(f (x), g(xp ))
has positive degree and that we can choose d(x) ∈ Z[x] and primitive such
that there is a(x) ∈ Z[x] and b(x) ∈ Z[x] primitive polynomials, such that
f (x) = a(x) × d(x) and g(xp ) = b(x) × d(x).
(e) Map f (x) and g(xp ) is Fp [x] via the canonical morphism Z[x] → Fp [x], prove
that an irreducible factor of d(x) in Fp [x] divides both f (x) and g(x) and
reach to the contradiction, so that we prove that φn (x) is irreducible. (Hint:
think about criterium characterizing polynomial with multiple roots in Fp .)
3. Let ξ be a primitive n-th root of unity, prove that we have an isomorphism of
groups:
Z ×
)
Gal(Q(ξ)/Q) ' ( nZ
a 7→ σa : ξ 7→ ξ a
1
4. Factoring φn (x) in Fp . Let p be a prime, m be an integer not divisible by p, and
e ≥ 1 integer. In Fp [x]
(a) Prove that
φpe m (x) =
((xm − 1)
(xm − 1)p
Q
pe−1
e
d|n,d<m
φpe d (x)
(b) Prove by induction on d < m, that
e
φpe m (x) = φm (x)φ(p ) = φm (x)(p−1)p
e−1
where φ denote the Euler totient function.
Exercise 2(?): 40 points
Let F ⊂ K be fields in C with [K : F ] = 2. Prove that K/F is a normal extension.
Exercise 3(?): 20 points
Let K1 , K2 be normal extensions of Q. Let M denote the minimal field containing K1 ∪ K2 . Prove that M is a normal extension of Q. [One approach: Write
K1 = Q(α1 , . . . , αr ) where the αi are all the roots of some p(x) ∈ Q[x] and do similarly
for K2 .]
Exercise 4(?): 40 points
Let F ⊂ L ⊂ K be fields in C. Assume that K/F is normal and that L/F is normal.
Let τ ∈ Gal(K, F ) and σ ∈ Gal(K, L). Let l ∈ L
1. Using a result shown in class argue that τ (l) ∈ L.
2. Show that (τ στ −1 )(l) = l.
3. From the above show that Gal(K, L) is a normal subgroup (get out those Algebra
I notes!) of Gal(K, F ). (Assume its already been shown that it is a subgroup.
You only need show the normal part.)
4. In the case F = Q, L = Q(ω), K = F (α, ω) (with α = 21/3 , ω = e2πi/3 as in our
standard example) give the groups Gal(K, L) and Gal(K, F ) explicitly in terms
of permutations of α, β = αω, γ = αω 2 .
2