INTEGRAL ELEMENTS AND EXTENSIONS
PETE L. CLARK
Recall that a complex number
α
is said to be an
algebraic integer
if
α
is the root
of a nonconstant monic polynomial with
Z
coefficients: i.e., if there exists an
n
and
integers
a
0
, . . . , a
n

1
such that
α
n
+
a
n

1
α
n

1
+
. . .
+
a
1
α
+
a
0
= 0
.
In order to prove the Quadratic Reciprocity Law, we used the following fact:
Proposition 1.
Let
n
be a positive integer and
ζ
n
a primitive
n
th root of unity.
Then every element of the ring
R
n
=
Z
[
ζ
n
]
is an algebraic integer.
We give two proofs here.
The first is a quick one, which however assumes the
following fact that one should learn in undergraduate algebra:
a subgroup of a
finitely generated abelian group is finitely generated.
One can deduce this from
the structure theory of modules over a PID, although it is in fact easier (if less
“undergraddy”) to use a little bit of the theory of Noetherian rings. Then we will
give a second proof, longer but selfcontained, of a much more general result.
1.
Proof of Proposition 1
Let
α
be any element of
R
n
=
Z
[
ζ
n
], and consider the subring
Z
[
α
] generated by
α
. Note that
R
n
is a finitely generated abelian group: indeed, it is generated by
1
, ζ
n
, . . . , ζ
n

1
n
. (With a bit more care, one can verify that
R
n
∼
=
Z
[
T
]
/
Φ
n
(
T
), so
that
R
n
∼
=
Z
ϕ
(
n
)
as an abelian group. But we don’t need this.)
Instead, we will use the fact that
Z
[
α
], being a subgroup of the finitely gener
ated abelian group
R
n
, is itself finitely generated: that is, there exists a finite set
of elements
a
1
, . . . , a
N
of
Z
[
α
] such that every element of
Z
[
α
] can be written as a
Z
linear combination of the
a
i
’s:
β
∈
Z
[
α
] =
⇒
β
=
r
1
a
1
+
. . .
+
r
N
a
N
, r
i
∈
Z
.
Now each element
a
i
∈
Z
[
α
] is, by definition, a polynomial in
α
with integer coef
ficients, say
a
i
=
f
i
(
α
). Let
t
be the maximum degree of these polynomials. We
may therefore write
β
=
α
t
+1
as
α
t
+1
=
r
1
f
1
(
α
) +
. . .
+
r
N
f
N
(
α
)
,
which shows that
α
satisfies the monic polynomial
T
t
+1

r
1
f
1
(
T
)

. . .

r
N
f
N
(
t
)
and is therefore an algebraic integer.
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 Spring '11
 Staff
 Algebra, Number Theory, Integers, Algebraic number theory, pete l. clark, algebraic integer

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