**§ ****6 ****Algebraic Numbers**

[ Algebraic Numbers ] If *q* is an algebraic equation whose coefficients are rational numbers

_{}

The root of , then *q* is called an algebraic number . After general division , *q* satisfies an algebraic equation with rounded coefficients , so an algebraic number can also be defined as "the root of an algebraic equation with rounded coefficients ".

If *f* ( *x* ) is an irreducible polynomial in the field of rational numbers , and , then *q* is called an algebraic number of degree *n* . Obviously , an algebraic number of first degree is a rational number ._{}

Algebraic numbers have the following properties :

1 ° The sum, difference, product, and quotient of two algebraic numbers ( with non-zero division ) are still algebraic numbers .

The roots of algebraic equations whose coefficients are algebraic numbers are still algebraic numbers .

[ Algebraic Integer ] If *q** is the root of an n* -th degree irreducible algebraic equation in which one coefficient is 1 and the other coefficients are rational integers , then *q* is called an *n* -th degree algebraic integer .

Algebraic integers have the following properties :

If the 1 ° algebraic integer is a rational number ( that is, a first-order algebraic integer ), it must be a rational integer .

2 ° The sum, difference and product of two algebraic integers are still algebraic integers .

3 ° The first coefficient is 1, and the roots of the algebraic equations whose other coefficients are algebraic integers are still algebraic integers .

4 ° If *q* is an algebraic number , it satisfies the rational integer coefficient equation

_{}

is an algebraic integer ._{}

5 ° If *q* is an algebraic integer of degree *n* , then the power of *q* can be expressed as _{}

_{}

where *i* is a non-negative integer *,* all of which are rational integers ._{}

6 ° If *q* is an algebraic number of degree *n* , then the power of *q* satisfies the equation _{}

_{}

where *i* is a non-negative integer *, which* is a rational integer ._{}

[ Unit number ] If both *q* and q are algebraic integers , then *q* is called a unit number ._{}

Unit numbers have the following properties :

The necessary and sufficient conditions for 1 ° *q* to be a unit number are : *q* is the root of the rational integer coefficient algebraic equation whose leading term is 1 and the constant term is 1 . _{}

2 ° The leading coefficient and constant term are all unit numbers , and the roots of algebraic equations whose other coefficients are algebraic integers are unit numbers .

[ algebraic extension ]

1 ° Single extension field , let *q* be an *n* -th algebraic number , then the form is

_{} ( Coefficients are rational numbers )

The totality of the numbers constitutes a field . It is called the *n* -time single extension field obtained by adding *q* to the rational number field **Q** , denoted as **Q** ( *q* ). If , then **Q** ( *q* ) is the addition, subtraction, and multiplication of the algebraic number *q* . , the largest set of numbers produced by division ( divisor non-zero ) ._{}

2 ° Finite extended field The field generated by the addition, subtraction, multiplication and division of a finite number of algebraic numbers ( the divisor is not zero ) is called the finite extended field on Q **,** denoted as _{}

**K** = **Q** ( )_{}

A finite extension field must be a single extension field , that is, there is an algebraic number *q* such that

**Q** ( ) = **Q** ( _{}*q* )

*The degree of q* is called the degreeof the finite extended field** Q** ()_{}.

[ Conjugate number ] Let *q* be an algebraic number of degree *n* , and *q* satisfies an irreducible polynomial of degree *n* on the rational number field* *

_{}

Note , and set the other *n* - 1 roots of the polynomial , then it is called the conjugate root of *q* ._{}_{}_{}

Any algebraic number *a*** Q** ( *q* ), then *a* can be uniquely represented as_{}

_{}
(1)

where is a rational number . Remember , then_{}_{}

_{}

is called the conjugate number of *a* .

[ Trace and Moment of Algebraic Numbers ] Let *a*** K** = **Q** ( *q* ), note , let it be the conjugate number of a *,* then respectively call_{}_{}_{}

_{}

_{}

is the trace and moment of the algebraic number *a* , which is defined as formula (1) ._{}

Note that the trace and moment here are for the field **K** , and the moment is also called the norm . Another definition of them is: Let the minimal polynomial of *a* ( the lowest degree irreducible multinomial with *a* as the root ) be* *

_{}

order , then_{}

_{}

Traces and moments have the following properties :

1 ° If *a* is an algebraic number , then the trace and moment of a are rational *numbers* .

2 ° If *a* is an algebraic integer , then the trace and moment of *a* are rational integers . If *a* is a non-zero algebraic integer , then . _{}

The necessary and sufficient conditions for the 3 ° algebraic integer *a* to be a unit number are : . _{}

4 ° *S* ( *a** + **b* ) *= S* ( *a* ) *+ S* ( *b* ) * *

* N* ( *a **b* ) *= N* ( *a* ) *N* ( *b* )

[ Basics and whole bases of algebraic number fields ]

1 ° basis Let **K** be an *n* -th algebraic extension field , which is a set of algebraic numbers in **K.** If any algebraic number *g*** in K** can be uniquely represented as _{}

_{}

is a rational number in the formula , then it is called a set of basis of **K.** Obviously , it is linearly independent in the rational number field ._{}_{}_{}

_{}The necessary and sufficient conditions for being a set of bases of domain **K** are :

_{}

where is the conjugate number , *j=* ._{}_{}_{}

If **K=Q** ( *q* ), then it is a set of basis of **K.**_{}

2 ° Integer base Let **K** be an *n* -th algebraic extension field , a set of algebraic integers in **K** , if any algebraic integer *g*** in K** can be uniquely represented as _{}

_{}

where is a rational integer , it is called a set of integral bases of **K.**_{}_{}

If a group uses_{}

_{}

is the smallest algebraic integer , then this set is a set of integral bases ._{}

[ Quadratic field ] Let *D* be a rational integer without square factor . Then **Q** ( ) _{}is a quadratic field . _{}Any algebraic number in Q ( ) can **be** expressed as

_{}

where *a* and *b* are both rational numbers .

Let *D* be a rational integer with no square factor .

_{}

Then 1, *w* are a set of integral bases of the quadratic field **Q** ( ) _{}.

In general , the *n* -th field **Q** may not necessarily be able to find the algebraic integer *w* , so that_{}

_{}

a set of integral bases that form **Q** ( *q * ) .

[ Gaussian field ] Let **Q** ( *i* ) be called the Gaussian field , which is a quadratic field ._{}

Any number in the Gaussian domain can be expressed as

_{}

In the formula, *a and b* are both rational numbers . When both *a and b* are rational integers , *a+bi* is called a Gaussian integer .

The Gaussian field has four unit numbers : , _{}all of which have moments of 1.

[ Cycloidal field ] Let *m* be a positive integer , and the field ** S** formed by adding all the roots of the polynomial to

There is an

If *q* is an *m* -th primitive unit root , so that it is also an *m* -th primitive unit root , there are ( *m* ) in total , where ( *m* ) is the Euler function ._{}_{}_{}

[ decomposition theorem ]

1 ° Divisibility If *a** and **b* are two algebraic integers , when they are still algebraic integers , then *b* is said to be divisible by *a* , denoted as . At this time, a is said *to* be a multiple of *b* , and *b* is *a* factor of a . _{}_{}

2 ° Associativity If two algebraic integers *a** and **b* differ only by a single factor , then *a* and *b* are said to be associative .

Obviously : (i) *a* combines with a *;* (ii) if *a* combines with *b* , then *b* combines with *a* ; (iii) if *a* combines with *b* , and *b* combines with *g* , then *a* and *g* combine Combine . * *

3 ° Indecomposable If the algebraic integer *a* in **K** has two other algebraic integers *b, gK **,** and **neither* of them is a single number , such that _{}

*a = **b **g*

*Then a* is said to be decomposable in the field **K** , otherwise it is said to be non-decomposable .

4 ° Decomposition Theorem Any algebraic integer in **K** can be decomposed into the product of non-decomposable algebraic integers .

If this decomposition is unique regardless of order and associativity , it is called unique decomposition .

The unique decomposition theorem of the Gaussian field holds .

The unique decomposition theorem in the quadratic domain holds , it is now known that there are

**Q** ( ), *D* = 2, 3, 5, 6, 7, 13, 17, 21, 29 _{} , etc.

Not all unique factorization theorems of quadratic fields hold , for example , the unique factorization theorem of **Q** ( ) _{}does not hold :

_{}

* Rational integers are integers in the usual sense,which are here to distinguish them from algebraic integers.

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