In the mathematical branch of algebraic number theory , the whole ring of an algebraic number field is the analogue of the ring of whole numbers in the case of the field of rational numbers . The elements of a wholeness ring are called algebraic integers , the set of all algebraic integers is the wholeness ring in the field of all algebraic numbers .
definition
Let it be an algebraic number field , i.e. H. a finite extension of the field of rational numbers . Then the wholeness ring of is defined as the whole closure of in , i.e. H. the subset of those that have an equation of form
K
{\ displaystyle K}
O
K
{\ displaystyle {\ mathcal {O}} _ {K}}
K
{\ displaystyle K}
Z
{\ displaystyle \ mathbb {Z}}
K
{\ displaystyle K}
x
∈
K
{\ displaystyle x \ in K}
x
n
+
c
n
-
1
x
n
-
1
+
...
+
c
1
x
+
c
0
=
0
{\ displaystyle x ^ {n} + c_ {n-1} x ^ {n-1} + \ ldots + c_ {1} x + c_ {0} = 0}
meet with . Note that the coefficient of (the leading coefficient of the polynomial ) must be 1. Such polynomials are called normalized . Without these restrictions you would get the whole body .
c
i
∈
Z
{\ displaystyle c_ {i} \ in \ mathbb {Z}}
x
n
{\ displaystyle x ^ {n}}
x
n
+
c
n
-
1
x
n
-
1
+
...
+
c
1
x
+
c
0
{\ displaystyle x ^ {n} + c_ {n-1} x ^ {n-1} + \ ldots + c_ {1} x + c_ {0}}
K
{\ displaystyle K}
An equivalent definition reads: The wholeness ring of is the maximum order in the sense of inclusion , the main order on .
K
{\ displaystyle K}
K
{\ displaystyle K}
properties
O
K
{\ displaystyle {\ mathcal {O}} _ {K}}
is a finitely generated, free module of rank .
Z
{\ displaystyle \ mathbb {Z}}
[
K
:
Q
]
{\ displaystyle [K \ colon \ mathbb {Q}]}
O
K
{\ displaystyle {\ mathcal {O}} _ {K}}
is a dedekind ring .
The unit group of is described by the Dirichlet theorem of units .
O
K
{\ displaystyle {\ mathcal {O}} _ {K}}
Examples
Is , so is the ring of Eisenstein numbers
K
=
Q
(
i
3
)
{\ displaystyle K = \ mathbb {Q} (\ mathrm {i} {\ sqrt {3}})}
O
K
{\ displaystyle {\ mathcal {O}} _ {K}}
u
+
v
⋅
-
1
+
i
3
2
{\ displaystyle u + v \ cdot {\ frac {-1+ \ mathrm {i} {\ sqrt {3}}} {2}}}
With
u
,
v
∈
Z
.
{\ displaystyle u, v \ in \ mathbb {Z}.}
Such a number is the zero of the polynomial
X
2
-
(
2
u
-
v
)
X
+
(
u
2
-
u
v
+
v
2
)
.
{\ displaystyle X ^ {2} - (2u-v) X + (u ^ {2} -uv + v ^ {2}).}
Conversely, satisfies the polynomial equation
x
=
a
+
b
i
3
∈
K
{\ displaystyle x = a + b \ mathrm {i} {\ sqrt {3}} \ in K}
x
2
+
p
x
+
q
=
0
{\ displaystyle x ^ {2} + px + q = 0}
With
p
,
q
∈
Z
,
{\ displaystyle p, q \ in \ mathbb {Z},}
so follows and . One can show that then and are integer, so is
p
=
-
2
a
{\ displaystyle p = -2a}
q
=
a
2
+
3
b
2
{\ displaystyle q = a ^ {2} + 3b ^ {2}}
a
+
b
{\ displaystyle a + b}
2
b
{\ displaystyle 2b}
x
=
(
a
+
b
)
+
2
b
⋅
-
1
+
i
3
2
{\ displaystyle x = (a + b) + 2b \ cdot {\ frac {-1+ \ mathrm {i} {\ sqrt {3}}} {2}}}
an Eisenstein number.
Is , then is the ring of whole Gaussian numbers .
K
=
Q
(
i
)
{\ displaystyle K = \ mathbb {Q} (\ mathrm {i})}
O
K
{\ displaystyle {\ mathcal {O}} _ {K}}
Z
[
i
]
{\ displaystyle \ mathbb {Z} [\ mathrm {i}]}
In general, a wholeness basis looks like this for the wholeness ring of (with whole and square-free):
Q
(
d
)
{\ displaystyle \ mathbb {Q} ({\ sqrt {d}})}
d
{\ displaystyle d}
{
1
,
d
}
{\ displaystyle \ {1, {\ sqrt {d}} \}}
if congruent 2 or 3 mod 4
d
{\ displaystyle d}
{
1
,
1
+
d
2
}
{\ displaystyle \ {1, {\ frac {1 + {\ sqrt {d}}} {2}} \}}
if congruent 1 mod 4
d
{\ displaystyle d}
If a primitive -th root of unity denotes , the wholeness ring of the -th body of the circle is the same .
ζ
{\ displaystyle \ zeta}
n
{\ displaystyle n}
n
{\ displaystyle n}
Q
(
ζ
)
{\ displaystyle \ mathbb {Q} (\ zeta)}
Z
[
ζ
]
{\ displaystyle \ mathbb {Z} [\ zeta]}
See also
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