Prove that
the cokernel of the map
given by multiplication by
is isomorphic to
.
Show that the minimal polynomial of an algebraic number
is unique.
Which of the following rings have infinitely
many prime ideals?
The integers
.
The ring
of polynomials over
.
The quotient ring
.
The ring
of polynomials over the ring
.
The quotient ring
, for a fixed positive integer .
The rational numbers
.
The polynomial ring
in three variables.
Which of the following numbers are algebraic integers?
The number
.
The number
.
The value of the infinite sum
.
The number , where is a root of
.
Prove that
is not noetherian.
Let
.
Is an algebraic integer?
Explicitly write down the minimal polynomial of
as an element of
.
Which are the following rings are orders in the given
number field.
The ring
in the number field
.
The ring
in the number field
.
The ring
in the number field
.
The ring
in the number field
.
We showed in the text (see
Proposition 3.1.3) that
is integrally
closed in its field of fractions. Prove that and every nonzero
prime ideal of
is maximal. Thus
is not a Dedekind
domain only because it is not noetherian.
Let be a field.
Prove that the polynomial ring
is a Dedekind domain.
Is
a Dedekind domain?
Prove that every finite integral domain is a field.
Give an example of two ideals in a
commutative ring whose product is not equal to the set
.
Suppose is a principal ideal domain.
Is it always the case that
for all ideals in ?
Is the set
of rational numbers with
denominator a power of a fractional ideal?
Suppose you had the choice of the following two jobs^{20.1}:
Job 1
Starting with an annual salary of $1000,
and a $200 increase every year.
Job 2
Starting with a semiannual salary of $500,
and an increase of $50 every 6 months.
In all other respects, the two jobs are exactly alike.
Which is the better offer (after the first year)?
Write a Sage program that creates a table showing how
much money you will receive at the end of each year for
each job. (Of course you could easily do this by hand - the
point is to get familiar with Sage.)
Let be the ring of integers of a number field.
Let denote the abelian group of fractional ideals of .
Prove that is torsion free.
Prove that is not finitely generated.
Prove that is countable.
Conclude that if and are number fields, then there
exists some (non-canonical) isomorphism of groups
.
From basic definitions, find the rings of integers of the fields
and
.
In this problem, you will give an example to illustrate the
failure of unique factorization in the ring of integers of
.
Give an element
that factors in two distinct
ways into irreducible elements.
Observe explicitly that the factors uniquely, i.e.,
the two distinct factorization in the previous part of this problem
do not lead to two distinct factorization of the ideal
into prime ideals.
Factor the ideal as a product of primes
in the ring of integers of
. You're allowed
to use a computer, as long as you show the commands you use.
Let be the ring of integers of a number field ,
and let
be a prime number. What is the cardinality
of in terms of and
,
where is the ideal of generated by ?
Give an example of each of the following, with proof:
A non-principal ideal in a ring.
A module that is not finitely generated.
The ring of integers of a number field of degree .
An order in the ring of integers of a number field of degree .
The matrix on of left multiplication by an element of ,
where is a degree number field.
An integral domain that is not integrally closed in its field of fractions.
A Dedekind domain with finite cardinality.
A fractional ideal of the ring of integers of a number
field that is not an integral ideal.
Let
be a homomorphism of (commutative) rings.
Prove that if
is an ideal, then
is an ideal of .
Prove moreover that if is prime, then
is
also prime.
Let be the ring of integers of a number field.
The Zariski topology on the set
of all prime ideals
of has closed sets the sets of the form
where varies through all ideals of , and
means that
.
Prove that the collection of closed sets of the
form is a topology on .
Let be the subset of nonzero prime ideals of , with
the induced topology. Use
unique factorization of ideals to prove
that the closed subsets of are exactly the finite subsets
of along with the set .
Prove that the conclusion of (a) is still true if is replaced
by an order in , i.e., a subring that has finite
index in as a
-module.
Explicitly factor the ideals generated by each of , , and in
the ring of integers of
. (Thus you'll factor separate
ideals as products of prime ideals.)
You may assume that the ring of integers of
is
, but do not simply
use a computer command to do the factorizations.
Let
,where
is a primitive
th root of unity. Note that has ring of integers
.
Factor , , , , , and in the ring
of integers . You may use a computer.
For , find a conjectural
relationship between the number of prime ideal factors of
and the order of the reduction of in
.
Compute the minimal polynomial
of
.
Reinterpret your conjecture as a conjecture that
relates the degrees of the irreducible factors of
to
the order of modulo . Does your conjecture
remind you of quadratic reciprocity?
Find by hand and with proof
the ring of integers of each of the following two fields:
,
.
Find the ring of integers of
, where
using a computer.
Let be a prime. Let be the ring of integers of a
number field , and suppose is such that
is finite and coprime to . Let be the
minimal polynomial of . We proved in class
that if the reduction
of factors
as
where the are distinct irreducible polynomials in
, then the primes appearing
in the factorization of are the ideals
.
In class, we did not prove that the exponents of these primes in the factorization
of are the . Prove this.
Let ,
, and
as elements of
.
Prove that the ideals , , and
are coprime in pairs.
Compute
.
Find a single element in
that is congruent to modulo ,
for each .
Find an example of a field of degree at least such that the ring
of integers of is not of the form
for any .
Let
be a prime ideal of , and suppose that
is a finite field of characteristic
. Prove that there is
an element
such that
. This
justifies why we can represent prime ideals of as pairs
, as is done in SAGE. (More generally, if is an
ideal of , we can choose one of the elements of to be any nonzero element of .)
(*) Give an example of an order in the ring of integers of
a number field and an ideal such that cannot be generated by
elements as an ideal. Does the Chinese Remainder Theorem hold
in ? [The (*) means that this problem is more difficult than
usual.]
For each of the following three fields, determining if there is
an order of discriminant contained in its ring of integers:
and
any extension of
of degree . [Hint: for the last one,
apply the exact form of our theorem about finiteness of class groups
to the unit ideal to show that the discriminant of a degree
field must be large.]
Prove that the quantity in our theorem about finiteness
of the class group can be taken to be
, as follows (adapted from [SD01, pg. 19]):
Let be the set of elements
such that
Prove that is convex and that ,
where
[Hint: For convexity, use the triangle inequality and
that for
, we have
for
. In polar coordinates this last inequality
is
which is trivial. That
follows from the inequality
between the arithmetic and geometric means.
Transforming pairs
from Cartesian to polar coordinates,
show also that
, where
and
is given by
(
),
(
) and
Prove that
and deduce by induction that
Let vary through all number fields. What torsion
subgroups
actually occur?
If
, we say that
has rank . Let vary through all number fields.
What ranks actually occur?
Let vary through all number fields such that the
group of units of is a finite group. What finite groups
actually occur?
Let
.
Show that and .
Find explicit generators for the group of
units .
Draw an illustration of the log map
, including the hyperplane
and the lattice in the hyperplane
spanned by the image of .
Let be a number field. Prove that if and only
if ramifies in . (Note: This fact is proved in many
books.)
Using Zorn's lemma, show that there are homomorphisms
with finite image that are not continuous, since
they do not factor through the Galois group of any finite Galois
extension. [Hint: The extension
is an extension of
with Galois group
.
The index-two open subgroups of correspond to the quadratic
extensions of
. However, Zorn's lemma implies that contains
many index-two subgroups that do not correspond to quadratic
extensions of
.]
Give an example of a finite nontrivial Galois extension of
and a prime ideal
such that
.
Give an example of a finite nontrivial Galois extension of
and a prime ideal
such that
has order .
Give an example of a finite Galois extension of
and a prime ideal
such that
is not a normal
subgroup of
.
Give an example of a finite Galois extension of
and a prime ideal
such that
is not a normal
subgroup of
.
Let by the symmetric group on three symbols, which
has order .
Observe that
, where is the dihedral group
of order , which is the group of symmetries of an equilateral
triangle.
Let be the number field
,
where
is a nontrivial cube root of unity. Show
that is a Galois extension with Galois group isomorphic to .
We thus obtain a -dimensional irreducible complex
Galois representation
Compute a representative matrix of and the characteristic polynomial
of for
.
Look up the Riemann-Roch theorem in a book on algebraic curves.
Write it down in your own words.
Let be an elliptic curve over a field .
Use the Riemann-Roch theorem to deduce that the natural map
is an isomorphism.
Suppose is a finite group and is a finite -module.
Prove that for any , the group is a torsion abelian group of
exponent dividing the order of .
Let
and let be the group of units of
, which is a module over the group
. Compute the
cohomology groups and . (You shouldn't use
a computer, except maybe to determine .)
Let
and let be the class group of
, which is a module over the Galois group
.
Determine and .
Let be the elliptic curve
. Let
be the group of points of order dividing on . Let
be the mod Galois representation associated to .
Find the fixed field of
.
Is
surjective?
Find the group
.
Which primes are ramified in ?
Let be an inertia group above , which is one
of the ramified primes. Determine explicitly
for your choice of . What is the characteristic polynomial
of acting on .
What is the characteristic polynomial of acting
on ?
Let be a number field. Prove that there is a finite
set of primes of such that
all
is a prinicipal ideal domain. The condition
means that in the prime ideal factorization of the fractional ideal
, we have that
occurs to a nonnegative power.
Let and a positive integer. Prove that
is unramified outside the primes that divide and the
norm of . This means that if
is a prime of , and
is coprime to
, then the prime factorization of
involves no primes with exponent bigger than .
Write down a proof of Hilbert's Theorem 90, formulated
as the statement that for any number field , we have
Let be any field. Prove that the only nontrivial valuations
on which are trivial on are equivalent to the valuation
(13.3.3) or (13.3.4) of page .
A field with the topology induced by a valuation is
a topological field, i.e., the operations sum, product,
and reciprocal are continuous.
Give an example of a non-archimedean valuation on a field that
is not discrete.
Prove that the field
of -adic numbers is
uncountable.
Prove that the polynomial
has all its roots in
, and find the -adic valuations
of each of these roots. (You might need to use
Hensel's lemma, which we don't discuss in detail
in this book. See [Cas67, App. C].)
In this problem you will compute an example of weak
approximation, like I did in the Example 14.3.3. Let
, let
be the -adic absolute value, let
be the -adic absolute value, and let
be the usual archimedean absolute value. Find an
element
such that
, where ,
, and
.
Prove that has a cube root in
using the following
strategy (this is a special case of Hensel's Lemma, which you can
read about in an appendix to Cassel's article).
Show that there is an element
such that
.
Suppose .
Use induction to show that if
and
, then there exists
such
that
.
(Hint: Show that there is an integer such that
.)
Conclude that has a cube root in
.
Compute the first digits of the -adic expansions of the following
rational numbers:
the 4 square roots of $41$
Let be an integer. Prove that the series
converges in
.
Prove that has a cube root in
using the following strategy (this
is a special case of ``Hensel's Lemma'').
Show that there is
such that
.
Suppose .
Use induction to show that if
and
, then there exists
such
that
.
(Hint: Show that there is an integer such that
.)
Conclude that has a cube root in
.
Let be an integer.
Prove that
is equipped with a natural ring structure.
If is prime, prove that
is a field.
Let and be distinct primes. Prove that
.
Is
isomorphic to either of
or
?
Prove that every finite extension of
``comes from'' an extension of
, in the following sense.
Given an irreducible polynomial
there exists an
irreducible polynomial
such that the fields
and
are isomorphic. [Hint: Choose each
coefficient of to be sufficiently close to the corresponding
coefficient of , then use Hensel's lemma to show that has a
root in
.]
Find the -adic expansion to precision 4 of each root of the following polynomial over
:
Your solution should conclude with three expressions of the form
Find the normalized Haar measure of the following subset of
:
Find the normalized Haar measure of the subset
of
.
Suppose that is a finite extension of
and
is a finite extension of
, with and assume
that and have the same degree. Prove that
there is a polynomial
such that
and
. [Hint: Combine your solution to 13 with the weak approximation theorem.]
Prove that the ring defined in Section 9 really is the tensor
product of and , i.e., that it satisfies the defining universal
mapping property for tensor products. Part of this problem is for you
to look up a functorial definition of tensor product.
Find a zero divisor pair in
.
Is
a field?
Is
a field?
Suppose denotes a primitive th root of unity. For
any prime , consider the tensor product
. Find a simple
formula for the number of fields appearing in the
decomposition of the tensor product
.
To get full credit on this problem your formula must be correct, but
you do not have to prove that it is correct.
Suppose
and
are
equivalent norms on a finite-dimensional vector space
over a field (with valuation
).
Carefully prove that the topology induced by
is the same as that induced by
.
Suppose and are number fields (i.e., finite
extensions of
). Is it possible for the tensor
product
to contain a nilpotent element?
(A nonzero element in a ring is nilpotent if
there exists such that .)
Let be the number field
.
In how many ways does the -adic valuation
on
extend to a valuation on ?
Let
be a valuation on that extends
.
Let be the completion of with respect to .
What is the residue class field
of ?
Prove that the product formula holds for
similar to the
proof we gave in class using Ostrowski's theorem for
. You may
use the analogue of Ostrowski's theorem for
, which you had
on a previous homework assignment. (Don't give a measure-theoretic
proof.)
Prove Theorem 18.3.5, that ``The global field
is discrete in and the quotient
of additive
groups is compact in the quotient topology.'' in the case when
is a finite extension of
, where
is a finite field.