18.786: Topics in Algebraic Number Theory
This course was the home page for the MIT class
18.786 (Topics in Algebraic Number Theory), for the spring 2006 semester.
This class was intended as a first course in algebraic number theory for
students familiar with elementary number theory and undergraduate abstract
algebra. Topics covered include number fields, class numbers,
Dirichlet's units theorem, cyclotomic fields, local fields, valuations,
decomposition and inertia groups, ramification, basic analytic methods,
and basic class field theory.
Quick information
Instructor: Kiran Kedlaya
Grader: Ruochuan Liu
Lectures: TR 1112:30, room 2102
Office hours: Wed 12:301:30, or by appointment, or dropin
Homeworks due: Thursdays in lecture
On this site
Assignments
Corrections
These corrections have now been made on the PDF files for archival purposes;
I kept the corrections listed for the benefit of students who took the course
and may have copies of the uncorrected originals.

PS 1: in
problem 1, the bound "r < f" should be "r < g"; in problem 5,
(p1)/2 should be (p1)/4; and in problem 6, 121 should be 301.

PS 2: problem 2
is actually two problems run together by mistake (call them 2a and 2b); in
the second one, assume further that R is a domain. Hint
on problem 6: suppose that R has a maximal ideal M of height greater than 1,
and then construct an Mprimary ideal which is not a power of M.

PS 4: problem 8 might
become easier after we do completions; if you want to postpone it until
then, go ahead. Correction on that problem: R_2 should be finite,
not just integral, over R_1; but you probably figured that's what I meant!

PS 5:
7(d) is
incorrect as written; you should indeed get S_4 as the answer to (c). To get
a correct version of (d), replace "3torsion" by "5torsion" in (a). Also,
in 7(e), the prime 2 should be replaced by 3 (or by 5, if you changed (d)).

PS 6:
in
problem 5, take the splitting field of the polynomial P rather than the field
generated by a single root (or take Q[sqrt{D}] and adjoin a root of
P to that; the two formulations are equivalent). In
problem 7, the definition of the different should say that
Trace_{L/K}(xy) is in o_K, not equal to zero. Clarification: in problem 3,
the multiplicative set "generated by S" consists of elements which
generate ideals whose prime factors all belong to S.

PS 7:
in 1(a), both occurrences of x + zeta^i * p should be x + zeta^i y.
In 1(b), the minus sign should be a plus sign.

PS 8:
in 9(d), the second sentence means to say "Suppose that the number of faces of
R with one vertex in A and one in B is odd." And "faces" is what Monsky
uses to describe what you might prefer to call "edges".

PS 9:
in problem 8, I gave the wrong topology. The topology should be
the one with a basis given by a product of open sets, one in each K_v,
with all but finitely many equal to the ring of integers in K_v.
Also, problem 4 is not quite correct at stated. It's fine if the polynomial
has distinct roots; otherwise, the roots might only lie in an extension of
K.
 PS 10:
in problem 1(c), the map is to U_i/U_{i+1}, not U_{i+1}/U_{i+2}.
In problem 3(a), assume that the action is faithful (that is, the group
is a subgroup of the symmetric group on the things being moved).
In problem 4(d), K/Q is supposed to be abelian, not just Galois (sorry).
In problem 5, the containment should be in Q_p(zeta_{p^{e_p} m})
for some m coprime to p (the m is missing on the original).
In problem 7b, "Galois" should be "Galois and abelian".

PS 11: in problem 2(b), 5^i should be 2^i
and the sum runs to i=4 rather than i=5;
in problem 3, there was an alpha that
should have been an alpha_j; in problem 7, I forgot to specify that n = deg(P)
and that n is prime (though not necessarily equal to p);
in problem 11, t^p should be t^(p).