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63
Paper 4, Section II
20F Number Fields
Explain what is meant by an integral basis for a number field. Splitting
√ into the
cases d ≡ 1 (mod 4) and d ≡ 2, 3 (mod 4), find an integral basis for K = Q( d) where
d 6= 0, 1 is a square-free integer. Justify your answer.
√
Find the fundamental unit in Q( 13). Determine all integer solutions to the
equation x2 + xy − 3y 2 = 17.
Paper 2, Section II
20F Number Fields
(i) Show that each prime ideal in a number field K divides a unique rational prime p.
Define the ramification index and residue class degree of such an ideal. State and prove a
formula relating these numbers, for all prime ideals dividing a given rational prime p, to
the degree of K over Q.
Qn−1
(ii) Show that if ζn is a primitive nth root of unity then j=1
(1 − ζnj ) = n. Deduce
that if n = pq, where p and q are distinct primes, then 1 − ζn is a unit in Z[ζn ].
(iii) Show that if K = Q(ζp ) where p is prime, then any prime ideal of K dividing
p has ramification index at least p − 1. Deduce that [K : Q] = p − 1.
Paper 1, Section II
20F Number Fields
State a result involving the discriminant of a number field that implies that the
class group is finite.
√
Use Dedekind’s
√ theorem to√factor 2, 3, 5 and 7 into prime ideals in K = Q( −34).
By factoring 1 + −34 and 4 + −34, or otherwise, prove that the class group of K is
cyclic, and determine its order.
Part II, 2014
List of Questions
[TURN OVER
63
Paper 4, Section II
20H Number Fields
State Dedekind’s
criterion. Use it to factor the primes up to 5 in the ring of integers
√
OK of K = Q( 65). Show that every ideal in OK of norm 10 is principal, and compute
the class group of K.
Paper 2, Section II
20H Number Fields
(i) State Dirichlet’s unit theorem.
(ii) Let K be a number field. Show that if every conjugate of α ∈ OK has absolute value
at most 1 then α is either zero or a root of unity.
√
√
(iii) Let k = Q( 3) and K = Q(ζ) where ζ = eiπ/6 = (i + 3)/2. Compute NK/k (1 + ζ).
Show that
∗
OK
= {(1 + ζ)m u : 0 6 m 6 11, u ∈ Ok∗ }.
Hence or otherwise find fundamental units for k and K.
[You may assume that the only roots of unity in K are powers of ζ.]
Paper 1, Section II
20H Number Fields
Let f ∈ Z[X] be a monic irreducible polynomial of degree n. Let K = Q(α), where
α is a root of f .
(i) Show that if disc(f ) is square-free then OK = Z[α].
(ii) In the case f (X) = X 3 − 3X − 25 find the minimal polynomial of β = 3/(1 − α) and
hence compute the discriminant of K. What is the index of Z[α] in OK ?
[Recall that the discriminant of X 3 + pX + q is −4p3 − 27q 2 .]
Part II, 2013
List of Questions
[TURN OVER
63
Paper 4, Section II
20F Number Fields
√ √
Let K = Q( p, q) where p and q are distinct primes with p ≡ q ≡ 3 (mod 4). By
computing the relative traces TrK/k (θ) where k runs through the three quadratic subfields
of K, show that the algebraic integers θ in K have the form
1
1
√
√ √
θ = (a + b p) + (c + d p) q ,
2
2
where a, b, c, d are rational integers. Show further that if c and d are both even then a and
b are both even. Hence prove that an integral basis for K is
√
√
√
p+ q
1 + pq
√
p,
1,
,
.
2
2
Calculate the discriminant of K.
Paper 2, Section II
20F Number Fields
Let K = Q(α) where α is a root of X 2 − X + 12 = 0. Factor the elements 2, 3, α
and α + 2 as products of prime ideals in OK . Hence compute the class group of K.
Show that the equation y 2 + y = 3(x5 − 4) has no integer solutions.
Paper 1, Section II
20F Number Fields
Let K be a number field, and OK its ring of integers. Write down a characterisation
of the units in OK in terms of the norm. Without assuming Dirichlet’s units theorem,
prove that for K a quadratic field the quotient of the unit group
√ by {±1} is cyclic√(i.e.
generated by one element). Find a generator in the cases K = Q( −3) and K = Q( 11).
Determine all integer solutions of the equation x2 − 11y 2 = n for n = −1, 5, 14.
Part II, 2012
List of Questions
[TURN OVER
61
Paper 1, Section II
20F Number Fields
√
Calculate the class group for the field K = Q( −17).
[You may use any general theorem, provided that you state it accurately.]
Find all solutions in Z of the equation y 2 = x5 − 17.
Paper 2, Section II
20F Number Fields
(i) Suppose that d > 1 is a square-free
integer. Describe, with justification, the ring of
√
integers in the field K = Q( d).
(ii) Show that Q(21/3 ) = Q(41/3 ) and that Z[41/3 ] is not the ring of integers in this field.
Paper 4, Section II
20F Number Fields
(i) Prove that the ring of integers OK in a real quadratic field K contains a non-trivial
unit. Any general results about lattices and convex bodies may be assumed.
(ii) State the general version of Dirichlet’s unit theorem.
√
√
(iii) Show that for K = Q( 7), 8 + 3 7 is a fundamental unit in OK .
[You may not use results about continued fractions unless you prove them.]
Part II, 2011
List of Questions
[TURN OVER
60
Paper 1, Section II
20G Number Fields
Suppose that m is a square-free positive integer, m > 5 , m 6≡ 1 (mod 4) . Show
√
that, if the class number of K = Q( −m ) is prime to 3 , then x3 = y 2 + m has at most
two solutions in integers. Assume the m is even.
Paper 2, Section II
20G Number Fields
√
Calculate the class group of the field Q( −14 ) .
Paper 4, Section II
20G Number Fields
Suppose that α is a zero of x 3 − x + 3 and that K = Q(α) . Show that [K : Q] = 3.
Show that OK , the ring of integers in K, is OK = Z [α] .
[You may quote any general theorem that you wish, provided that you state it clearly.
Note that the discriminant of x 3 + px + q is −4 p3 − 27q 2 .]
Part II, 2010
List of Questions
60
Paper 1, Section II
20H Number Fields
Suppose that K is a number field with ring of integers OK .
(i) Suppose that M is a sub-Z-module of OK of finite index r and that M is
closed under multiplication. Define the discriminant of M and of OK , and show that
disc(M ) = r 2 disc(OK ).
(ii) Describe OK when K = Q[X]/(X 3 + 2X + 1).
[You may assume that the polynomial X 3 + pX + q has discriminant −4p3 − 27q 2 .]
(iii) Suppose that f, g ∈ Z[X] are monic quadratic polynomials with equal discriminant d, and that d ∈
/ {0, 1} is square-free. Show that Z[X]/(f ) is isomorphic to Z[X]/(g).
[Hint: Classify quadratic fields in terms of discriminants.]
Paper 2, Section II
20H Number Fields
Suppose that K is a number field of degree n = r + 2s, where K has exactly r real
embeddings.
(i) Taking for granted the fact that there is a constant CK such that every integral
ideal I of OK has a non-zero element x such that |N (x)| 6 CK N (I), deduce that the class
group of K is finite.
√
(ii) Compute the class group of Q( −21), given that you can take
s
4
n!
CK =
|DK |1/2 ,
π
nn
where DK is the discriminant of K.
(iii) Find all integer solutions of y 2 = x3 −21.
Paper 4, Section II
20H Number Fields
Suppose that K is a number field of degree n = r + 2s, where K has exactly r real
embeddings.
Show that the group of units in OK is a finitely generated abelian group of rank at
most r + s − 1. Identify the torsion subgroup in terms of roots of unity.
[General results about discrete subgroups of a Euclidean real vector space may be used
without proof, provided that they are stated clearly.]
√
Find all the roots of unity in Q( 11).
Part II, 2009
List of Questions
45
1/II/20G
Number Fields
(a) Define the ideal class group of an algebraic number field K. State a result
involving the discriminant of K that implies that the ideal class group is finite.
√
(b) Put K = Q(ω), where ω = 12 (1 + −23), and let OK be the ring of integers
of K. Show that OK = Z + Zω. Factorise the ideals [2] and [3] in OK into prime ideals.
Verify that the equation of ideals
[2, ω][3, ω] = [ω]
holds. Hence prove that K has class number 3.
2/II/20G
Number Fields
(a)
√ Factorise the ideals [2], [3] and [5] in the ring of integers OK of the field
K = Q( 30). Using Minkowski’s bound
n!
nn
s
p
4
|dK |,
π
determine the ideal class group of K.
[Hint: it might be helpful to notice that
3
2
= NK/Q (α) for some α ∈ K.]
(b) Find the fundamental unit of K and determine all solutions of the equations
x − 30y 2 = ±5 in integers x, y ∈ Z. Prove that there are in fact no solutions of
x2 − 30y 2 = 5 in integers x, y ∈ Z.
2
4/II/20G
Number Fields
(a) Explain what is meant by an integral
√ basis of an algebraic number field. Specify
such a basis for the quadratic field k = Q( 2).
√
(b) Let K = Q(α) with α = 4 2, a fourth root of 2. Write an element θ of K as
θ = a + bα + cα2 + dα3
with a, b, c, d ∈ Q. By computing the relative traces TK/k (θ) and TK/k (αθ), show that if
θ is an algebraic integer of K, then 2a, 2b, 2c and 4d are rational integers. By further
computing the relative norm NK/k (θ), show that
a2 + 2c2 − 4bd and 2ac − b2 − 2d2
are rational integers. Deduce that 1, α, α2 , α3 is an integral basis of K.
Part II
2008
43
1/II/20H
Number Fields
√
Let K = Q( −26).
√
(a) Show that OK = Z[ −26] and that the discriminant dK is equal to −104.
(b) Show that 2 ramifies in OK by showing that [2] = p22 , and
√ that p2 is not a principal
¯
ideal. Show further that [3] = p3 p3 with p3 √
= [3, 1 − −26]. Deduce that neither
p3 nor p23 is a principal ideal, but p33 = [1 − −26].
(c) Show that 5 splits in OK by showing that [5] = p5 p¯5 , and that
NK/Q (2 +
√
−26) = 30.
Deduce that p2 p3 p5 has trivial class in the ideal class group of K. Conclude that
the ideal class group of K is cyclic of order six.
√
[You may use the fact that π2 104 ≈ 6.492.]
2/II/20H
Number Fields
√
√
Let K = Q( 10) and put ε = 3 + 10.
(a) Show that 2, 3 and ε + 1 are irreducible elements in OK . Deduce from the equation
6 = 2 · 3 = (ε + 1)(¯
ε + 1)
that OK is not a principal ideal domain.
(b) Put p2 = [2, ε + 1] and p3 = [3, ε + 1]. Show that
[2] = p22 ,
[3] = p3 p¯3 ,
p2 p3 = [ε + 1] ,
p2 p¯3 = [ε − 1] .
Deduce that K has class number 2.
(c) Show that ε is the fundamental unit of K. Hence prove that all solutions in integers
x, y of the equation x2 − 10y 2 = 6 are given by
x+
Part II
2007
√
10y = ±εn (ε + (−1)n ) , n = 0, 1, 2, . . .
44
4/II/20H
Number Fields
Let K be a finite extension of Q and let O = OK be its ring of integers. We will
assume that O = Z[θ] for some θ ∈ O. The minimal polynomial of θ will be denoted by
g. For a prime number p let
g¯(X) = g¯1 (X)e1 · . . . · g¯r (X)er
be the decomposition of g¯(X) = g(X)+pZ[X] ∈ (Z/pZ)[X] into distinct irreducible monic
factors g¯i (X) ∈ (Z/pZ)[X]. Let gi (X) ∈ Z[X] be a polynomial whose reduction modulo p
is g¯i (X). Show that
pi = [p, gi (θ)] , i = 1, . . . , r ,
are the prime ideals of O containing p, that these are pairwise different, and
[p] = pe11 · . . . · perr .
Part II
2007
41
1/II/20G
Number Fields
Let α, β, γ denote the zeros of the polynomial x3 − nx − 1, where n is an integer.
The discriminant of the polynomial is defined as
∆ = ∆(1, α, α2 ) = (α − β)2 (β − γ)2 (γ − α)2 .
Prove that, if ∆ is square-free, then 1, α, α2 is an integral basis for k = Q(α).
By verifying that
α(α − β)(α − γ) = 2nα + 3
and further that the field norm of the expression on the left is −∆, or otherwise, show
that ∆ = 4n3 − 27. Hence prove that, when n = 1 and n = 2, an integral basis for k is
1, α, α2 .
2/II/20G
Number Fields
√
√
Let K = Q( 26) and let ε = 5 + 26. By Dedekind’s theorem, or otherwise, show
that the ideal equations
2 = [2, ε + 1]2 ,
5 = [5, ε + 1][5, ε − 1],
[ε + 1] = [2, ε + 1][5, ε + 1]
hold in K. Deduce that K has class number 2.
Show that ε is the fundamental unit in K. Hence verify that all solutions in integers
x, y of the equation x2 − 26y 2 = ±10 are given by
x+
√
26y = ±εn (ε ± 1)
(n = 0, ±1, ±2, . . .) .
[It may be assumed that the Minkowski constant for K is 12 .]
4/II/20G
Number Fields
Let ζ = e2πi/5 and let K = Q(ζ). Show that the discriminant of K is 125. Hence
prove that the ideals in K are all principal.
Verify that (1 − ζ n )/(1 − ζ) is a unit in K for each integer n with 1 6 n 6 4.
Deduce that 5/(1 − ζ)4 is a unit in K. Hence show that the ideal [1 − ζ] is prime and
totally ramified in K. Indicate briefly why there are no other ramified prime ideals in K.
[It can be assumed that ζ, ζ 2 , ζ 3 , ζ 4 is an integral basis for K and that the Minkowski
constant for K is 3/(2π 2 ).]
Part II
2006
39
1/II/20G
Number Fields
√ √
Let K = Q( 2, p) where p is a prime with p ≡ 3 (mod 4). By computing the
relative traces TrK/k (θ) where k runs through the three quadratic subfields of K, show
that the algebraic integers θ in K have the form
√
√ √
θ = 12 (a + b p) + 12 (c + d p) 2 ,
where a, b, c, d are rational integers. By further computing the relative norm NK/k (θ)
√
where k = Q( 2), show that 4 divides
a2 + pb2 − 2 c2 + pd2
and 2 ab − 2cd .
Deduce that a and b are even and c ≡ d (mod 2). Hence verify that an integral basis for
K is
√
√ √
√
1,
2,
p, 12 1 + p 2.
2/II/20G
Number Fields
√
√
√
Show that ε = (3 + 7)/(3 − 7) is a unit in k√= Q( 7). Show further that 2 is
the square of the principal ideal in k generated by 3 + 7.
Assuming that the Minkowski constant for k is 21 , deduce that k has class number 1.
Assuming further that ε is the fundamental unit in k, show that the complete
solution in integers x, y of the equation x2 − 7y 2 = 2 is given by
x+
√
7y = ±εn (3 +
√
7)
(n = 0, ±1, ±2, . . .).
Calculate the particular solution in positive integers x, y when n = 1.
4/II/20G
Number Fields
State Dedekind’s theorem on the factorisation of rational primes into prime ideals.
A rational prime is said to ramify totally in a field with degree
√ n if it is the n-th
power of a prime ideal in the field. Show that, in the quadratic field Q( d) with d a squarefree integer, a rational prime ramifies totally if and only if it divides the discriminant of
the field.
Verify that the same holds in the cyclotomic
field Q(ζ), where ζ = e2πi/q with q an
√
3
odd prime, and also in the cubic field Q( 2).
[The cases d ≡ 2, 3 (mod 4) and d ≡ 1 (mod 4) for the quadratic field should be carefully
distinguished. It can be assumed
that Q(ζ) has√ a basis
1, ζ, . . . , ζ q−2 and discriminant
√
√
3
3
3
(−1)(q−1)/2 q q−1 , and that Q( 2) has a basis 1, 2, ( 2)2 and discriminant −108.]
Part II
2005