Lecture 2 Notes: Plane Conics
2.1
Plane conics
A conic is a plane projective curve of degree 2. Such a
curve has the form
C=k : ax2 + by2 + cz 2 + dxy +
exz + f yz
with a; b; c; d; e; f 2 k. Assuming the
characteristic of k is not 2, we can make d = e
= f
Lecture 7 Notes: Field Norms
7.1
o
f
g
.
Field norms and traces
Let L=K be a _nite _eld extension of
degree n = [L : K]. Then L is an ndimensional K-vector space, and each 2 L
determines a linear operator T : L ! L
T
h
i
s
corresponding to multiplication
r
i
n
g
Lecture 4 Notes: Inverse Limits
4.1
Inverse limits
Z
De_nition 4.1. An inverse system is a
sequence of objects (e.g. sets/groups/rings)
(An ) together with a sequence of morphisms
p
h
a
s
(e.g. functions/homomorphisms) (fn )
_
! An+1
A2
! An
! _
!
Lecture 3 Notes: Quadratic Reciprocity
3.1
Quadratic reciprocity
Recall that for each odd prime p the Legendre symbol
( p ) is de_ned as
8if a is a
_
nonzero
_ quadratic
residue
<
1 modulo p; if
p a is zero
modulo p;
= otherwise:
:
0
1
The Legendre symbol
Lecture 8 Notes: Completions of Q
We already know that R is the completion of Q with respect to its archimedean absolute
value j j1 . Now we consider the completion of Q with respect to any of its nonarchimedean
absolute values j jp .
Theorem 8.1. The com
Lecture 5 Notes: P_ Adic Numbers
As a fraction _eld, the elements of Qp are by de_nition all pairs (a; b) 2 Zp , typically
written as a=b, modulo the equivalence relation a=b _ c=d whenever ad = bc. But we can
represent elements of Qp more explicitly by e
Lecture 11 Notes: Quadratic forms over Qp
The Hasse-Minkowski theorem reduces the problem of determining whether a quadratic
form f over Q represents 0 to the problem of determining whether f represents zero over
Qp for all p _ 1. At _rst glance this migh
Lecture 12 Notes: Field extensions
Before beginning our introduction to algebraic geometry we recall some standard facts
about _eld extensions. Most of these should be familiar to you and can be found in any
standard introductory algebra text, such as We
Lecture 10 Notes: The Hilbert symbol
De_nition 10.1. For a; b 2 Qp the Hilbert symbol (a; b)p is de_ned by
ax2 + by 2 = 1 has a solution in Qp ;
(
(a; b)p =
1
otherwise:
It is clear from the de_nition that the Hilbert symbol is symmetric, and that it only
Lecture 9 Notes: Quadratic forms
We assume throughout k is a _eld of characteristic dierent from 2.
De_nition 9.1. The four equivalent de_nitions below all de_ne a quadratic form on k.
1. A homogeneous quadratic polynomial f 2 k[x1 ; : : : ; xn ].
2. Asso