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Algebraic integer ring

Special commutative domain

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Ring of algebraic integers Integer ring , is a special commutative integer ring, Algebraic number field All algebraic integers O in K K An integer ring called K, where K is O K Let L ⊃ K be two Number field , then O L Yes O K At L Integral closure , OL is also a finitely generated OK module, OK is Dai De Gold Ring , which ideal Can be uniquely (regardless of order) decomposed into Prime ideal Product of, O K Is a unique factorial ring if and only if O K yes Principal ideal ring , which is also equivalent to that the number of ideal classes of K is 1. From the Theorem of Module Structure on the Dadkin Ring（ Yuval Steinitz (Steinitz, E.) (1912) - Kaplansky, I. (1952)) L ≌O n-1 K ⊕ J, where n=[L ∶ K], J is an ideal in K, and the ideal class of J is uniquely determined by L and K. In particular, when J is the main ideal (for example, it is always the case when the number of ideal classes of K is 1), there is O L ≌O n K , that is, there is ω one ，ω two ，…，ω n ∈O L Make O L =O K ω one ⊕…⊕O K ω n^[1] 。

Chinese name: Algebraic integer ring
Foreign name: ring of algebraic integers
Discipline: mathematics

Alias: Integer ring
Properties: A Special Commutative Domain
Field: Number Theory (Algebraic Number Theory)
Definition: All Algebraic Integers O in Algebraic Number Field K K Integer ring called K

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Basic Introduction

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Ring of algebraic integers Integer ring , is a special commutative domain, and all algebraic integers in algebraic number field K are O K An integer ring called K, where K is O K Business domain of, set

Is two number fields, then O L Yes O K In the whole closure of L, O L Is also a finitely generated O K Die, O K Is Dade Golden Ring, whose ideal can be uniquely (regardless of order) decomposed into the product of its prime ideal, O K Is a unique factorial ring if and only if O K Is a principal ideal ring, which is also equivalent to that the number of ideal classes of K is 1. It is known from the structure theorem of modules on the Dedekin ring (Steinitz, E. (1912) - Kaplansky, I. (1952)),

, where n=[L ∶ K], J is an ideal in K, and the ideal class of J is uniquely determined by L and K. In particular, when J is the main ideal (for example, it is always the case when the number of ideal classes of K is 1), there are

, that is, there is ω one ，ω two ，…，ω n ∈O L Make O L =O K ω one ⊕…⊕O K ω n^[1] 。

Algebraic integer

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Algebraic integer integer ， Algebraic number One of. It is a generalization of rational integers (i.e. natural numbers, zeros and their opposite numbers). Set α as complex If there is a polynomial f (x) whose coefficient is the first rational integer (that is, the coefficient of the highest degree term is 1) so that f (α)=0, then α is called an algebraic integer. If the constant term of the above f (x) is ± 1, α is called the unit, all integers form a commutative ring I, and its quotient field (or fractional field) is the field A formed by all algebraic numbers, and the unit is the reversible element in ring I. A notable feature of algebraic integers is that they may not be able to perform unique irreducible factorization, for example,

As a result, the concept of ideal is introduced, and the concept of integer is also extended to the ordinary arithmetic field F. If S is an assignment set of F, the elements in the intersection of the assignment rings in S are called S integers^[1] 。

Correlation theorem

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Definition 1 Algebraic number

Is called Algebraic integer (referred to as integer for short), if there is a coefficient belonging to the first polynomial of Z

, making

^[2] 。

Lemma 1 set up

Is an algebraic number because

Minimal polynomial on Q, then

If and only if it is an integer

。

System 1 Only rational integers in Q are (algebraic) integers.

Theorem 1 with

Represent secondary field

(d is a rational integer without square factor)

When,

； When d=1 (mod 4),

, where

。

For the quadratic field K, you can directly verify the set of integers found in Theorem 1

It's actually a subring of K. For any number field K, Dedekind proved the set of integers of K

It is also a subring of K. In other words, if

and

Are integers in K, then

and

It is also an integer. This is an extraordinary thing（

Obviously

Imagine how to prove

It is also an integer). We need to give other characterizations of integers.

Theorem 2 about

, the following conditions are equivalent to each other:

(1)

Is an (algebraic) integer;

(2) Ring

The additive group of is finitely generated;

(3)

Is an element of a nonzero subring R of C, and the additive group of R is finitely generated;

(4) Existence of finitely generated nonzero additive subgroups

bring

。

Theorem 3 if

and

Are integers, then

It is also an integer. In particular, it is a set of all integers in the number field K

Is a subring of K.

Algebraic integer ring: take

It is called number field K Integer ring The integer ring of rational number field Q is Z, and the integer ring of quadratic field has been given by Theorem 1. Now let's give the cyclotomic field

Integer ring of.

Theorem 4 Cyclotomic domain K=

The integer ring of is

^[2] 。

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