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integral basis of quadratic field∗
pahio†
2013-03-22 0:51:21
√
Let m be a squarefree integer 6= 1. All numbers of the quadratic field Q( m)
may be written in the form
√
j+k m
,
(1)
α=
l
where j, k, l are integers with
gcd(j, k, l) = 1 and l > 0. Then α (and its
√
j−k m
0
algebraic conjugate α =
) satisfy the equation
l
x2 + px + q = 0,
(2)
where
p=−
2j
,
l
q=
j 2 − k2 m
.
l2
(3)
We will find out when the number (1) is an algebraic integer, i.e. when the
coefficients p and q are rational integers.
Naturally, p and q are integers always when l = 1. We suppose now that
l > 1. The latter of the equations (3) says that q can be integer only when
(gcd(j, l))2 = gcd(j 2 , l2 ) | k 2 m
(see divisibility in rings). Because gcd(j, k, l) = 1, we have by Euclid’s lemma
that gcd(j, l) | m. Since m is squarefree, we infer that
gcd(j, l) = 1.
(4)
In order that also p were an integer, the former of the equations (3) implies that
l = 2.
So, by the latter of the equations (3), 4 | j 2 − k 2 m, i.e.
k2 m ≡ j 2
(mod 4).
∗ hIntegralBasisOfQuadraticFieldi
(5)
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1
Since by (4), gcd(j, 2) = 1, the integer j has to be odd. In order that (5) would
be valid, also k must be odd. Therefore, j 2 ≡ 1 (mod 4) and k 2 ≡ 1 (mod 4),
and thus (5) changes to
m≡1
(mod 4).
(6)
If we conversely assume (6) and that j, k are odd and l = 2, then (5) is true,
p, q are integers and accordingly (1) is an algebraic integer.
We have now obtained the following result:
√
• When m 6≡ 1 (mod 4), the integers of the field Q( m) are
√
a+b m
where a, b are arbitrary rational integers;
√
• when m ≡ 1 (mod 4), in addition to the numbers a + b m, also the
numbers
√
j+k m
,
2
with j, k arbitrary odd integers, are integers of the field.
Then, it may be easily inferred the
Theorem. If we denote
(
√
1+ m
when m ≡ 1 (mod 4),
2
ω :=
√
m when m 6≡ 1 (mod 4),
√
then any integer of the quadratic field Q( m) may be expressed in the form
a + bω,
where a and b are uniquely determined rational integers. Conversely, every
number of this form is an integer of the field. One says that 1 and ω form an
integral basis of the field.
References
[1] K. Väisälä: Lukuteorian ja korkeamman algebran alkeet. Tiedekirjasto No.
17. Kustannusosakeyhtiö Otava, Helsinki (1950).
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