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ALGEBRAIC
NUMBER
THEORY
LECTURE
1
SUPPLEMENTARY
NOTES
Material covered: Sections
1.1 through 1.3 of textbook.
1.
Section
1.1
Recall that
to
an integral domain
A
we
can associate
its field of fractions
K
= Frac(
A
) =
{
a
b
:
b
=
negationslash
0
}
. More
formally,
K
=
{
(
a,b
) :
a,b
∈
A,b
=
negationslash
0
}
c
/
∼
,
where
∼
is the equivalence relation (
a,b
)
∼
(
c,d
) iff
ad
−
bc
= 0 (i.e.
“
a
=
”).
b
d
A general
fractional
ideal
f
of
A
is a subset
of
K
= Frac(
A
) such that
(1)
af
∈
f
∀
a
∈
A,f
∈
f
(2)
f
1
+
f
2
∈
f
∀
f
1
,f
2
∈
f
(3)
∃
c
∈
A
such that
c
f
is
an ideal
of
A
.
A
nonexample of
a
fractional
ideal
is the
set
K
:
it
satisfies the
first
two
properties but
not the third one in general:
the problem is that
we have inverted
“too
many”
elements.
Example.
Some principal ideal domains (PIDs):
(1)
Z
is a PID (any
nonzero ideal is a subgroup,
so is generated by a smallest
positive element).
(2)
For
a
field
k
, the ring
of univariate
polynomials
k
[
X
]
is a PID
(take
a
lowest
degree element).
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 Spring '10
 AbhinavKumar
 Algebra, Number Theory, Fractions, Ring, Prime number, xa, Principal ideal domain

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