Find invertible integer matrices and such
that is in Smith normal form.
What is the group structure of
the cokernel of the map
defined by multiplication by
?
Let be the abelian group generated by with relatoins
and . Find a product of cyclic groups that is
isomorphic to .
Prove that each of the following rings have infinitely
many prime ideals:
The integers
. [Hint: Euclid gave a famous proof of this long ago.]
The ring
of polynomials over
.
The ring
of polynomials over
.
The ring
of all algebraic integers. [Hint: Use Zorn's
lemma, which implies that every ideal is contained in a maximal
ideal. See, e.g., Prop 1.12 on page 589 of Artin's Algebra.]
(This problem was on the graduate qualifying exam on Tuesday.) Let
denote the subset of all elements of
that satisfy a
monic polynomial with coefficients in the ring
of integers. We
proved in class that
is a ring.
Show that the ideals and
in
are
distinct.
Prove that
is not Noetherian.
Show that neither
nor
is a unique factorization domain. [Hint: Consider
the factorization into irreducible elements of in the first case
and in the second. A nonzero element in a ring is an irreducible element if it is not a unit and if whenever , then
one of or is a unit.]
Find the ring of integers of each of the following number
fields:
,
, and
.
Do not use a computer for the first two.
Find the discriminants of the rings of integers of the numbers
fields in the previous problem. (Do not use a computer.)
Let be a finite integral domain. Prove that is a field.
[Hint: Show that if is a nonzero element, then has an inverse
by considering powers of .]
Suppose
is a tower of number fields
and let
be a field embedding of into
that fixes elementwise. Show that extends in exactly
ways to a field embedding
.
Suppose and are principal ideals in a ring .
Show that the set
is an ideal.
Give an example of ideals and in the polynomial
ring
in two variables such that
is not an ideal.
Your example illustrates why it is necessary to define the product
of two ideals to be the ideal generated by
.
Give an example of a ring of integers of a number
field, and ideals and such that
is not an ideal.
Let be a field. Prove that is a Dedekind domain.
(Problem 1.12 from Swinnerton-Dyer) Let be an indeterminate.
Show that the ring
is Noetherian and integrally closed in its
field of fractions, but is not a Dedekind domain.
Use to write each of the following (fractional) ideals as a
product of explicitly given prime ideals:
The ideal in
.
The ideals and in the ring of integers of
, where is a root of the irreducible polynomial
. (The field
is called the th cyclotomic field.)
The principal fractional ideal in
.
Suppose is an order in the ring of integers of a
number field. (Recall that an order is a subring of finite index in
.) For each of the following questions, either explain why the
answer is yes for any possible order in any , or find one
specific counterexample:
Is necessarily Noetherian?
Is necessarily integrally closed in its field of fractions?
Is every nonzero prime ideal of necessarily maximal?
Is it always possible to write every ideal of uniquely as a
product of prime ideals of ?
Let be the ring of integers of a number field . Prove
that the group of fractional ideals of , under multiplication is
(non-canonically) isomorphic to the group of positive rational numbers
under multiplication.
Suppose is a number field of degree . Prove that
for some .
Prove that if and are two number fields
of degree and
then
.
(*) Does there exist a number field of degree such that
for all ? If so, give an explicit example.
Let be the quintic number field generated by a root of
. Draw a diagram (be creative) that illustrates
the factorization of every prime
, with , in .
(Problem 1.9 in Swinnerton-Dyer) Show that the only solutions
to
are given by
, as follows.
Factor the equation
in the number field
, which has
class number . Show that if is an integer solution then the
ideal
must be the cube of an ideal, and hence
; thus
.
Suppose and are ideals in the ring of integers of
a number field . Does
? Prove or give a
counterexample.
Let be the ring of integers
,
and let
and
be integral ideals of .
Find an element of that is congruent to
modulo and is congruent to
modulo .
What is the cardinality of
?
Find an element such that is coprime
to .
Let be the ring of integers of a number field
, and suppose has exactly complex embeddings. Prove
that the sign of
is .
(*) Suppose is an order in the ring of integers of a
number field. Is every ideal in necessarily generated by two
elements? (Answer: No. Challenge: Given an example.)
Find representative ideals for each element of the class group
of
. Illustrate how to use the Minkowski bound
to prove that your list of representatives is complete.
Suppose is an order in the ring of integers of a
number field. Is every ideal in necessarily generated by two
elements?
Let be a number field of degree with pairs
of complex conjugate embeddings. Prove that
Do the exercise on page 19 of Swinnerton-Dyer, which shows
that the quantity in the finiteness of class group theorem
can be taken to be
.
Let denote a root of and let
.
Show that
and that has class number (don't
just read this off from the output of the commands MaximalOrder and ClassNumber). [Hint: consider the square
factors of the discriminant of and show that
is an algebra integer if and only
if , , and are all even.]
If is a closed, bounded, convex, symmetric set in
with
, for some positive integer , show that
contains at least nonzero points in
.
Prove that any finite subgroup of the multiplicative
group of a field is cyclic.
For a given number field , which seems more difficult for
to compute, the class groups or explicit generators for the
group of units? It is very difficult (but not impossible) to not get
full credit on this problem. Play around with some examples, see what
seems more difficult, and justify your response with examples.
(This problem might be annoying to do using the web page, since
it kills your job after 30 seconds. Feel free to request a
binary of from me, or an account on MECCAH (Mathematics Extreme
Computation Cluster at Harvard).)
Prove that there is no number field such that
.
Is there a number field such that
?
Prove that the rank of is unbounded as varies over all
number fields.
Let
.
Show that and .
Find explicitly generators for the group of
units of (you can use for this).
Draw an illustration of the log map
, including the hyperplane
and the lattice in the hyperplane
spanned by the image of .
Find the group of units of
as an abstract
group as a function of . (I.e., find the number of cyclic
factors and the size of the torsion subgroup. You do not have
to find explicit generators!)
Let
, where
is a root .
Show that .
Find explicitly the log embedding of
into a -dimensional hyperplane in
, and
draw a picture.
Prove that if is a quadratic field and the torsion subgroup
of has order bigger than , then
or
.
A is a real algebraic integer, greater than
1, with the property that all of its conjugates lie on or within the
unit circle, and at least one conjugate lies on the unit circle. By
any method (including ``google''), give two examples of Salem numbers.
Let
and let be a number field. Show that
.
A totally real number field is a number field in which all embeddings
into
have image in
. Prove there are totally real number
fields of degree , for every prime . [Hint: Let
denote a primitive th root of unity. For , show that
is totally real of degree
. Now
prove that
can be made divisible by any prime.]
Give an example of a number field
and a prime such
that the in the factorization of are not all the same.
Let be a number field. Give the ``simplest''
proof you can think of that there are only finitely many
primes that ramify (i.e., have some ) in .
[The meaning of ``simplest'' is a matter of taste.]
Give examples to show that for
a Galois extension, the
quantity can be arbirarily large and can be arbitrarily large.
Suppose
is Galois and is a prime such that
is also prime (i.e., is inert in ). Show
that
is a cyclic group.
(Problem 7, page 116, from Marcus Number Fields) For
each of the following, find
a prime and quadratic extensions and of
that illustrates the assertion:
The prime can be totally ramified in and without
being totally ramified in .
The fields and can each contain unique primes lying over
while does not.
The prime can be inert in and without being inert in .
The residue field extensions of
can be trivial
for and without being trivial for .
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
.
Let
.
Show that is Galois over
, compute the Galois
group of , and compute
.
Let be any field. Prove that the only nontrivial valuations
on which are trivial on are equivalent to the valuation
(15.3.3) or (15.3.4) 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 16.3.3. 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 59 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 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 20.3.5, 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.