Chapter 7
Introducing Algebraic Number
Theory
(Commutative Algebra 1)
The general theory of commutative rings is known as
commutative algebra
. The main
applications of this discipline are to algebraic number theory, to be discussed in this
chapter, and algebraic geometry, to be introduced in Chapter 8.
Techniques of abstract algebra have been applied to problems in number theory for
a long time, notably in the effort to prove Fermat’s Last Theorem. As an introductory
example, we will sketch a problem for which an algebraic approach works very well. If
p
is an odd prime and
p
≡
1 mod 4, we will prove that
p
is the sum of two squares, that is,
p
can be expressed as
x
2
+
y
2
where
x
and
y
are integers. Since
p
−
1
2
is even, it follows
that 1 is a quadratic residue (that is, a square) mod
p
. [Pair each of the numbers 2,3,
...
,
p
−
2 with its multiplicative inverse mod
p
and pair 1 with
p
−
1
≡ −
1 mod
p
. The
product of the numbers 1 through
p
−
1 is, mod
p
,
1
×
2
× · · · ×
p
−
1
2
× −
1
× −
2
× · · · × −
p
−
1
2
and therefore
(
p
−
1
2
)
!
2
≡ −
1 mod
p
.]
If
−
1
≡
x
2
mod
p
, then
p
divides
x
2
+ 1. Now we enter the ring of Gaussian integers
and factor
x
2
+ 1 as (
x
+
i
)(
x
−
i
). Since
p
can divide neither factor, it follows that
p
is
not prime in
Z
[
i
], so we can write
p
=
αβ
where neither
α
nor
β
is a unit.
Define the
norm
of
γ
=
a
+
bi
as
N
(
γ
) =
a
2
+
b
2
. Then
N
(
γ
) = 1 iff
γ
=
±
1 or
±
i
iff
γ
is a unit. (See Section 2.1, Problem 5.) Thus
p
2
=
N
(
p
) =
N
(
α
)
N
(
β
) with
N
(
α
)
>
1 and
N
(
β
)
>
1
,
so
N
(
α
) =
N
(
β
) =
p
. If
α
=
x
+
iy
, then
p
=
x
2
+
y
2
.
1
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CHAPTER 7. INTRODUCING ALGEBRAIC NUMBER THEORY
Conversely, if
p
is an odd prime and
p
=
x
2
+
y
2
, then
p
is congruent to 1 mod 4. (If
x
is even, then
x
2
≡
0 mod 4, and if
x
is odd, then
x
2
≡
1 mod 4. We cannot have
x
and
y
both even or both odd, since
p
is odd.)
It is natural to conjecture that we can identify those primes that can be represented as
x
2
+

d

y
2
, where
d
is a negative integer, by working in the ring
Z
[
√
d
]. But the Gaussian
integers (
d
=
−
1) form a Euclidean domain, in particular a unique factorization domain.
On the other hand, unique factorization fails for
d
≤ −
3 (Section 2.7, Problem 7), so the
above argument collapses. [Recall from (2.6.4) that in a UFD, an element
p
that is not
prime must be reducible.] Diﬃculties of this sort led Kummer to invent “ideal numbers”,
which later became ideals at the hands of Dedekind. We will see that although a ring of
algebraic integers need not be a UFD, unique factorization of ideals will always hold.
7.1
Integral Extensions
If
E/F
is a field extension and
α
∈
E
, then
α
is algebraic over
F
iff
α
is a root of a
polynomial with coeﬃcients in
F
. We can assume if we like that the polynomial is monic,
and this turns out to be crucial in generalizing the idea to ring extensions.
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 Spring '11
 sdd
 Algebra, Number Theory, Ring, Algebraic number theory, Ring theory, Principal ideal domain

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