(September 22, 2011)
Number theory exercises 02
Paul Garrett [email protected]
http:/
/www.math.umn.edu/ garrett/
Due Fri, 30 Sept 2011, preferably as PDF emailed to me.
[number theory 02.1] Show that the ideal norm and Galois norm agree on Z[i]. That
Garrett 09-19-2011
1
Continuing the pre/review of the simple (!?) case.
Examples: Riemanns formula
log p = X (b + 1) lim
T
|Im ()|<T
pm <X
X
+
n1
X 2n
2n
Gauss Quadratic Reciprocity:
q
p
2
p
q
= (1)
(p1)(q 1)
4
2
Continuing: factorization of Dedekind zet
Garrett 09-16-2011
1
Continuing the pre/review of the simple (!?) case.
So far, we have sketched the connection between prime numbers,
and zeros of the zeta function, given by Riemanns formula
log p = X (b + 1) lim
T
|Im ()|<T
pm <X
X
+
n1
X 2n
2n
A dier
Garrett 09-14-2011
1
Continuing the review of the simple (!?) case of number theory
over :
So far, we have sketched the connection between prime numbers,
and zeros of the zeta function, given by Riemanns formula
pm <X
log p = X (b + 1) lim
T
|Im ()|<T
X
(September 22, 2011)
Algebraic Number Theory Exercises-discussion 01
Paul Garrett [email protected]
http:/
/www.math.umn.edu/ garrett/
[number theory 01.1] Prove the Euler product expansion of the zeta function, namely, for Re (s) > 1
1
=
ns
n=1
p
prim
Garrett 09-09-2011
1
Continuing to review the simple case (haha!) of number theory
over Z:
Another example of the possibly-suprising application of othe
things to number theory.
Riemanns explicit formula
More interesting than a Prime Number Theorem is the
Garrett 09-21-2011
1
Continuing the pre/review of the simple (!?) case.
Continuing: factorization of Dedekind zeta-functions into Dirichlet
L-functions, equivalently, behavior of primes in extensions. So far,
[i] (s)
= (s) L(s, )
(p) =
1
p2
[ 2] (s)
= (s)
Garrett 09-23-2011
1
Continuing the pre/review of the simple (!?) case. Some
themes so far:
Riemanns explicit formula connects complex zeros of meromorphic
continuations of zeta functions (and L-functions) to tangible,
nitistic properties of primes.
Gauss
Garrett 09-26-2011
1
Continuing the pre/review .
Riemanns explicit formula: complex zeros of zeta functions (and
L-functions) versus properties of primes.
Gauss Quadratic Reciprocity via Gauss sums, which are Lagrange
resolvents for cyclotomic elds.
Facto
Garrett 09-30-2011
Continuing the pre/review .
Continuing: solving equations mod pn , p-adic numbers, Hensel.
Completions versus projective limits.
Mapping-property characterizations. unique up to unique
isomorphism.
Another forgotten point: not only are
(September 29, 2011)
Number theory exercises-discussion 02
Paul Garrett [email protected]
http:/
/www.math.umn.edu/ garrett/
Due Fri, 30 Sept 2011, preferably as PDF emailed to me.
[number theory 02.1] Show that the ideal norm and Galois norm agree on
(September 13, 2011)
Algebraic Number Theory Exercises 01
Paul Garrett [email protected]
http:/
/www.math.umn.edu/ garrett/
Due Wed, 21 Sept 2011, preferably as PDF emailed to me.
[number theory 01.1] Prove the Euler product expansion of the zeta funct
(September 29, 2011)
Number theory exercises 03
Paul Garrett [email protected]
http:/
/www.math.umn.edu/ garrett/
Due Mon, 10 Oct 2011, preferably as PDF emailed to me.
[number theory 03.1] Prove that 1 exists in Q5 .
[number theory 03.2] Prove that a
Garrett 09-12-2011
Review the simple (haha!) case of number theory over Z:
Continuing discussion of analytical properties of (s) relevant to
Riemanns Explicit Formula (von Mangoldts reformulation):
log p = X (b + 1) lim
pm <X
T
|Im ()|<T
X
+
n1
X 2n
2n
W