18.786 Midterm Exam
April 1, 2010
Solve two out of these three problems. Justify your answers: for instance, saying that a high-level gp
function (such as bnfclgp() outputs the answer you give is not enough justication. You may use gp
without justication
ALGEBRAIC NUMBER THEORY
LECTURE 10 NOTES
1. Section 5.1
Example (Rings of fractions). Let A be an integral domain.
(1) If S = A\cfw_0, we get the entire eld of fractions of A.
(2) If S = cfw_1, x, x2 , . . . , we get the lo calization Ax = cfw_a/xn : a A,
18.786 Problem Set 1 (due Thursday Feb 11)
1. Let a and b be positive integers such that ab + 1 divides a2 + b2 . Show that
(Hint: use descent in the sense of Fermat).
a2 +b2
ab+1
is a perfect square.
2. Give an alternative pro of of the structure theorem
ALGEBRAIC NUMBER THEORY
LECTURE 6 NOTES
Material covered: Class numbers of quadratic elds, Valuations, Completions
of elds.
1. Ideal class groups of quadratic fields
These are the ideal class groups of the Dedekind domains OK for quadratic
elds K . We alr
ALGEBRAIC NUMBER THEORY
LECTURE 7 NOTES
Material covered: Lo cal elds, Hensels lemma.
Remark. The non-archimedean topology: Recall that if K is a eld with a val
uation | |, then it also is a metric space with d(x, y ) = |x y |. The topology
has a basis of
ALGEBRAIC NUMBER THEORY
LECTURE 11 NOTES
First well prove the proposition from last time:
Proposition 1. Let A be a Dedekind domain with fraction eld K . Let L/K
be a nite separable extension, and B the integral closure of A in L. Assume B
is monogenic ov
ALGEBRAIC NUMBER THEORY
LECTURE 12 NOTES
1. Section 5.5
Note that (1)2 = ( 1 ) holds in characteristic 0 as well as characteristic q (set
p
w = e2i/p ), since it do esnt use any property of nite elds. It allows us to see
what the unique quadratic subeld o
ALGEBRAIC NUMBER THEORY
LECTURE 1 SUPPLEMENTARY NOTES
Material covered: Sections 1.1 through 1.3 of textbo ok.
1. Section 1.1
Recall that to an integral domain A we can asso ciate its eld of fractions
K = Frac(A) = cfw_ a : b = 0. More formally, K = cfw_(
ALGEBRAIC NUMBER THEORY
LECTURE 4 SUPPLEMENTARY NOTES
Material covered: Sections 2.6 through 2.9 of textbo ok.
I followed the bo ok pretty closely in this lecture, so only a few comments.
Wherever Samuel states a theorem with the assumption that a eld has
ALGEBRAIC NUMBER THEORY
LECTURE 2 SUPPLEMENTARY NOTES
Material covered: Sections 1.4 through 1.7 of textbo ok.
For the pro of of Theorem 1 of Section 1.5, a motivating example to keep in
mind is that of a lattice in Zn . The pro of using linear forms basi
ALGEBRAIC NUMBER THEORY
LECTURE 3 SUPPLEMENTARY NOTES
Material covered: Sections 2.1 through 2.5 of textbo ok.
1. Section 2.1
Pro of of Theorem 1, (b) (c): Once we have the system of equations,
(x I C )y = 0
where C = (aij ), we can multiply the matrix x
ALGEBRAIC NUMBER THEORY
LECTURE 5 SUPPLEMENTARY NOTES
Material covered: Chapter 3 of textbo ok.
1. Section 3.3
If A A and p is a prime ideal of A, then p = p A is a prime ideal of A ,
called the restriction of p to A .
Lemma 1. Suppose a1 , a2 , a3 are id
ALGEBRAIC NUMBER THEORY
LECTURE 8 NOTES
1. Section 4.1
We say a set S Rn is discrete if the topology induced on S is the discrete
topology. Check that this is equivalent to the denition in the bo ok (every
compact subset K of Rn intersects S in a nite set
ALGEBRAIC NUMBER THEORY
LECTURE 9 NOTES
1. Secton 4.4
Pro of of theorem 1: the bound on should be 2nr1 (1/2 )r2 |d|1/2 .
2. Section 4.6
Solving the Brahmagupta-Pell-Fermat eq tion:
ua
Write the continued fraction expansion of d as [a0 , a1 , . . . , ak ,
18.786 Problem Set 2 (due Thursday Feb 18)
pr
X
1
1. Show that for p a prime, r 1, the polynomial X pr1 is irreducible in Q[x]. Compute the ring of
1
integers of Q(pr ) and therefore the discriminant of this cyclotomic eld.
2. Let K be a number eld. Show