ALGEBRAIC NUMBER THEORY HOMEWORK
A note about assignments: The following are the exercises and assignments I suggest you try - however, not all exercises will be collected. I will mention during lecture which exercises I will be collecting. I will collect homework every other week.
Exercise 1. Show that Z[i] is an euclidean domain.
Exercise 2. Show that the only units in the ring of gaussian integers Z[i] are ±1 and ±i. Exercise 3. Show that the only units in Z[√−2] are ±1.
Exercise 4. Show that the elements 1 + i, 1 − i and 3, are irreducibles in Z[i].
Exercise 5. Show that the elements 2, 3, 1 +√−5, and 1 −√−5 are irreducibles in Z[√−5]. (Therefore, the equation 6 = 2 · 3 = (1 +√−5)(1 −√−5) shows that Z[√−5] is not a UFD.) Exercise 6. Let R be a unique factorization domain, and let γ, δ and τ be elements of R such that γ · δ = τn, for some natural number n ≥ 1.
(1) Suppose that gcd(γ, δ) = 1 (in other words, the only common divisors in R of γ and δ are units). Show that there are elements α, β ∈ R and units u and v in R, with u · v = 1, such that
γ = u · αn, and δ = v · βn.
(2) Suppose that gcd(γ, δ) = ρ ∈ R. Show that there are elements α, β, u and v ∈ R, such that u and v are only divisible by irreducibles dividing ρ, with u · v = ωn for
some ω ∈ R, such that
γ = u · αn, and δ = v · βn.
Exercise 7. Show that √2/3 is an algebraic number but not an algebraic integer. Find its minimal polynomial.
Exercise 8. Let d be a square-free integer. Show that if d ≡ 1 mod 4 then −1±
√ d 2 is an
algebraic integer.
Exercise 9. Let m ≥ 2 be a natural number and let ζm be a primitive mth root of unity.
Show that
Y
(ζmi − ζj
m) = (−1)
m−1mm
where the product is taken over all pairs (i, j) with 0 ≤ i, j ≤ m − 1 and i 6= j. (Hint: Factor xm− 1 and its derivative.)
Exercise 10. Let d be a square-free integer and let F = Q(√d). Let τ : F → C be a map defined by τ (s + t√d) = s − t√d, for any t, s ∈ Q. Show that τ is an embedding of F . Exercise 11. Let K/F be a finite extension of number fields and let α ∈ K. Let
m(x) = xn+ fn−1xn−1+ · · · + f1x + f0, fi ∈ F
be the minimal polynomial of α over F . Let σ be an embedding of K into C and let F0 = σ(F ). We define a polynomial mσ(x) ∈ F0[x] by:
mσ(x) = σ(m(x)) = xn+ σ(fn−1)xn−1+ · · · + σ(f1)x + σ(f0), σ(fi) ∈ F0 = σ(F ).
Show that F [x]/(m(x)) and F0[x]/(mσ(x)) are isomorphic.
Exercise 12. Let K/F be a finite extension of number fields and let α ∈ K. Define TrKF(α) = τ1(α) + τ2(α) + · · · + τd(α) = X τ ∈ΣK/F τ (σ) NKF(α) = τ1(α)τ2(α) · · · τd(α) = Y τ ∈ΣK/F τ (σ).
where ΣK/F is the set of all embeddings of K that fix F . Prove that TrKF(α) and N K
F(α) are
elements of F . Also, show that if α ∈ Z then TrKF(α) and N K
F(α) are algebraic integers as
well.
Exercise 13. Let d be a square-free integer, let F = Q(√d) and put OF = F ∩ Z.
(1) For each square-free d < 0, compute O×F, i.e. compute all the units in the ring of algebraic integers of F .
(2) Conclude that, if d < 0 then OF× is finite.
(3) Show that if K is a number field and OK× = (K ∩ Z)× is finite, then OK× only contains roots of unity, i.e., if u ∈ O×K then there is n ≥ 1 such that un = 1.
(4) Let K = Q(√2). Prove that O×K is infinite.
Exercise 14. Let n ≥ 3 and let ζn be a primitive nth root of unity.
(1) Show that the extension Q(ζn)/Q(ζn+ ζn−1) is a quadratic extension.
(2) Write down all the algebraic conjugates of θn = ζn+ ζn−1.
(3) Show that every conjugate of θn is a real number (we write Q(ζn)+ := Q(ζn+ ζn−1)
and we say that Q(ζn)+ is the maximal real subfield of Q(ζn)).
(4) What is the minimal polynomial of θn?
Exercise 15. Let ζ5and ζ7be primitive 5th and 7th roots of unity, respectively. For the fields
below, describe all their subfields by giving explicit generators (over Q) for each subfield. (1) Q(ζ5),
(2) Q(ζ7), and
Exercise 16. Let n, m ≥ 1, let qm(x) = xm − 1 and let Φn(x) be the nth cyclotomic
polynomial. Prove that the polynomial qm(x) is divisible by Φn(x) if and only if m is divisible
by n.
Exercise 17. Prove that the factorization of qm(x) = xm− 1 into irreducibles over Q[x] is
given by
xm− 1 = Y
n|m, n≥1
Φn(x).
Exercise 18. Let p ≥ 2 be a prime. Show that the polynomial q(x) = x
p− 1
x − 1 = x
p−1+ xp−2+ · · · + x2+ x + 1
is irreducible over Q[x]. (Hint: Use Eisenstein criterion on q(x + 1).)
Exercise 19. Let a ∈ Z≥1 be square-free, and let p be a prime. Show that the Galois closure (or normal closure) of Q(a1/p) is Q(a1/p, ζp), where ζp is a primitive p-th root of unity.
Exercise 20. Let K be a number field with [K : Q] = n, let m ≥ 1 be an integer and let α ∈ OK. Prove that
disc(1, α, α2, . . . , αn−1) = disc(1, (α + m), (α + m)2, . . . , (α + m)n−1). Exercise 21. Find an integral basis of OK, where K = Q(α) and α3+ 2α + 1 = 0.
Exercise 22. Let K = Q(α) where α is a root of p(x) = x3− x2− 2x − 8 = 0.
(1) Show that p(x) is irreducible over Q.
(2) Let β = (α2+ α)/2. Show that β is a root of q(x) = x3− 3x2− 10x − 8 = 0. Thus,
β ∈ OK.
(3) Show that disc(1, α, α2) = −4 · 503, and disc(1, α, β) = −503. (4) Find an integral basis for OK.
(5) Show that every integer τ ∈ OK satisfies disc(1, τ, τ2) is an even integer (in Z).
(6) Deduce that OK has no integral basis of the form Z[τ ].
Exercise 23. Find integral bases for the ring of integers of the following number fields: (1) K = Q(√2,√−3), and
(2) F = Q(√p,√q), where p, q are distinct primes congruent to 1 mod 4.
Exercise 24. Let K be a number field of degree n = r1+ 2r2 over Q, and let OK be its ring
of integers. Let ΣK = {σ1, . . . , σr1, τ1, τ1, . . . , τr2, τr2} be the set of all the real and complex
embeddings of K. Let µK be the group of roots of unity contained in K. Let u1, . . . , us, with
s = r1+ r2− 1, be generators for the quotient O×K/µK. Let M be the (r1+ r2) × (r1+ r2− 1)
matrix with the following vectors in Rr1+r2 as columns:
Ci = (log |σ1(ui)|, . . . , log |σr1(ui)|, log |τ1(ui)|, log |τ2(ui)|, . . . , log |τr2(ui)|).
Let Mi be the (r1 + r2 − 1) × (r1 + r2− 1) matrix that you obtain from M by omitting the
(1) Prove that | det(Mi)| = | det(Mj)| for all 1 ≤ i, j ≤ r1+ r2.
(2) Prove that the regulator RK is independent of the choice of generators u1, . . . , us.
Exercise 25. Let p ≥ 3 be a prime and let ζ = ζp be a primitive pth root of unity.
(1) Suppose that r and s are integers with gcd(p, rs) = 1. Show that u = 1 − ζ
r
1 − ζs
is a unit of Z[ζ].
(2) Let ζ = e2πi/p and α = eπi/p. Show that
1 − ζk = −2iαksin(kπ/p) for all k ∈ Z.
(3) Let α be as above, and let k ∈ Z. Show that 1 − ζk
1 − ζ = α
k−1sin(kπ/p)
sin(π/p) . (4) Show that αk−1 = ±ζh for some h ∈ Z.
(5) Show that if k is relatively prime to p then uk =
sin(kπ/p) sin(π/p) is an algebraic number and a unit in Z[ζ].
(6) Show that all the algebraic conjugates of uk are real numbers.
Exercise 26. Let K be a number field such that K/Q is a Galois extension, and let dK =
disc(K).
(1) Show that K contains the quadratic field Q(√dK).
(2) Let p > 2 be a prime, and let ζp be a primitive pth root of unity. Let K = Q(ζp).
Show that K contains the quadratic field Q(√±p). Moreover, K contains Q(√p) if and only if p ≡ 1 mod 4.
(3) Suppose that K/Q is Galois and of prime degree [K : Q] = p > 2. What can you say about dK?
Exercise 27. Let K = Q(√3
2) and let OK be its ring of integers. Determine the prime ideal
factorization of 7OK, 29OK and 31OK in prime ideals of OK.
Exercise 28. Let K = Q(α) where α is a root of p(x) = x3− x2− 2x − 8 = 0, and let O K be
the ring of integers of K. Assuming the results of Exercise 22, find primes p ∈ Z, and prime ideals ℘ ⊆ OK above p, such that the following hold (or explain why there are no examples):
(1) The ideal pZ remains prime in OK.
(2) The ideal pZ splits completely in OK.
(4) The inertial and ramification degree are f (℘|p) = e(℘|p) = 2. (5) The inertial and ramification degree are f (℘|p) = e(℘|p) = 1. (6) The ramification degree e(℘|p) = 3.
Exercise 29. Use Minkowski’s bound (or otherwise) to calculate the class number of the following cyclotomic fields:
Q(i), Q(ζ3), Q(ζ5), and Q(ζ7).
Exercise 30. Let K = Q(α) where α is a root of p(x) = x5 − x + 1. Show that the class
number of K equals 1.
Exercise 31. The goal of this exercise is to show that the class number of a quadratic imaginary field can be arbitrarily large. Let n > 1 and let p be a prime such that p ≡ 5 mod 12, and p > 3n. Let K = Q(√−p) and let O
K be the ring of integers of K.
(1) Show that the prime 3 splits in K. In other words, there are two distinct prime ideals ℘ and ℘0 of OK such that 3OK = ℘℘0.
(2) Suppose that the order of the class [℘] in the ideal class group Cl(K) is exactly m ≥ 1, i.e. ℘m is principal. Show that there exist u, v ∈ Z such that
u2 + pv2 = 3m.
(3) Show that if m ≤ n then v = 0 and m must be even.
(4) Show that if m ≤ n then ℘m = ℘m/2(℘0)m/2. Conclude that m > n. (Hint: Unique Factorization.)
(5) Show that the order of Cl(K) is greater than n.
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