The ring of integer p-adic numbers. Ring of integers
Federal Agency for Education
State educational institution of higher professional education
Vyatka State University for the Humanities
Faculty of Mathematics
Department of Mathematical Analysis and Methods
teaching mathematics
Final qualifying work
on the topic: Gauss ring of integers.
Completed:
5th year student
Faculty of Mathematics
Gnusov V.V.
___________________________
Scientific adviser:
senior lecturer of the department
algebra and geometry
Semenov A.N.
___________________________
Reviewer:
Candidate of Physics and Mathematics Sciences, Associate Professor
Department of Algebra and Geometry
Kovyazina E.M.
___________________________
Admitted to defense in the SAC
Head Department ________________ Vechtomov E.M.
« »________________
Dean of the faculty ___________________ Varankina V.I.
Introduction.
Ring of integer complex numbers
was discovered by Carl Gauss and named Gaussian after him.K. Gauss came to the idea of the possibility and necessity of expanding the concept of an integer in connection with the search for algorithms for solving comparisons of the second degree. He transferred the concept of an integer to numbers of the form
, where are arbitrary integers, and is the root of the equation On this set, K. Gauss was the first to construct a theory of divisibility, similar to the theory of divisibility of integers. He substantiated the validity of the basic properties of divisibility; showed that there are only four invertible elements in the ring of complex numbers: ; proved the validity of the theorem on division with a remainder, the theorem on the uniqueness of decomposition into prime factors; showed which prime natural numbers will remain prime in the ring; discovered the nature of simple integer complex numbers.The theory developed by K. Gauss, described in his work "Arithmetical Investigations", was a fundamental discovery for number theory and algebra.
The following goals were set for the thesis:
1. Develop the theory of divisibility in the ring of Gauss numbers.
2. Find out the nature of simple Gaussian numbers.
3. Show the application of Gaussian numbers in solving ordinary Diophantine problems.
CHAPTER 1. DIVISIBILITY IN THE RING OF GAUSS NUMBERS.
Consider the set of complex numbers. By analogy with the set of real numbers, a subset of integers can be distinguished in it. The set of numbers of the form
, where will be called complex integers or Gaussian numbers. It is easy to check that the axioms of the ring hold for this set. Thus, this set of complex numbers is a ring and is called ring of Gaussian integers . Let's denote it as , since it is an extension of the ring by the element: .Since the ring of Gaussian numbers is a subset of complex numbers, then some definitions and properties of complex numbers are valid for it. For example, for each Gaussian number
corresponds to a vector starting at point and ending at . Consequently, module Gaussian numbers are . Note that in the set under consideration, the submodule expression is always a non-negative integer. Therefore, in some cases it is more convenient to use the norm , that is, the square of the modulus. In this way . We can distinguish the following properties of the norm. For any Gaussian numbers, the following is true: (1) (2) (3) (4) (5) - the set of natural numbers, that is, positive integers.The validity of these properties is trivially checked using the module. In passing, we note that (2), (3), (5) are also valid for any complex numbers.
The ring of Gaussian numbers is a commutative ring without divisors 0, since it is a subring of the field of complex numbers. This implies the multiplicative contractibility of the ring
, i.e. (6)1.1 REVERSIBLE AND ALLOY ELEMENTS.
Let's see which Gaussian numbers will be reversible. Multiplication neutral is
. If a Gaussian number reversible , then, by definition, there exists such that . Passing to the norms, according to property 3, we obtain . But these norms are natural, therefore. Hence, by property 4, . Conversely, all elements of this set are invertible, since . Therefore, numbers with a norm equal to one will be reversible, that is, , .As you can see, not all Gaussian numbers will be reversible. Therefore, it is interesting to consider the issue of divisibility. As usual, we say that
is divided on if there exists such that . For any Gaussian numbers , as well as invertibles, the properties are true. (7) (8) (9) (10) , where (11) (12)(8), (9), (11), (12) are easily verified. The validity (7) follows from (2), and (10) follows from (6). Due to property (9), the elements of the set
Definition:
The sum and product of integer p-adic numbers defined by the sequences u are the integer p-adic numbers defined respectively by the sequences u.
To be sure of the correctness of this definition, we must prove that the sequences and determine some integers - adic numbers, and that these numbers depend only on, and not on, the choice of the sequences that define them. Both of these properties are proved by an obvious check.
Obviously, given the definition of actions on integer - adic numbers, they form a communicative ring containing the ring of integer rational numbers as a subring.
The divisibility of integer - adic numbers is defined in the same way as in any other ring: if there is such an integer - adic number that
To study the properties of division, it is important to know what are those integers - adic numbers, for which there are reciprocal integers - adic numbers. Such numbers are called unit divisors or units. We will call them - adic units.
Theorem 1:
An integer is an adic number defined by a sequence if and only if it is one when.
Proof:
Let be a unit, then there is such an integer - an adic number, that. If it is determined by a sequence, then the condition means that. In particular, and hence, Conversely, let From the condition it easily follows that, so. Therefore, for any n one can find such that the comparison is valid. Since and then. This means that the sequence determines some integer - an adic number. Comparisons show that, i.e. which is the unit.
From the proved theorem it follows that the integer is a rational number. Being considered as an element of the ring, if and only then is the unit when. If this condition is met, then the It follows that any rational integer b is divisible by such a in, i.e. that any rational number of the form b/a, where a and b are integers and, is contained in Rational numbers of this form are called -integers. They form an obvious ring. Our result can now be formulated as follows:
Consequence:
The ring of integer - adic numbers contains a subring isomorphic to the ring - of integer rational numbers.
Fractional p-adic numbers
Definition:
A fraction of the form, k >= 0 defines a fractional p-adic number or simply a p-adic number. Two fractions, and, determine the same p-adic number, if c.
The collection of all p-adic numbers is denoted p. It is easy to check that the operations of addition and multiplication continue from p to p and turn p into a field.
2.9. Theorem. Every p-adic number is uniquely represented in the form
where m is an integer and is the unit of the ring p .
2.10. Theorem. Any non-zero p-adic number can be uniquely represented in the form
Properties: The field of p-adic numbers contains the field of rational numbers. It is easy to prove that any integer p-adic number that is not a multiple of p is invertible in the ring p, and that is a multiple of p is uniquely written in the form where x is not a multiple of p and therefore is invertible, a. Therefore, any non-zero element of the field p can be written in the form where x is not a multiple of p, but m is any; if m is negative, then, based on the representation of integer p-adic numbers as a sequence of digits in the p-ary number system, we can write such a p-adic number as a sequence, that is, formally represent it as a p-ary fraction with a finite the number of digits after the decimal point, and possibly an infinite number of non-zero digits before the decimal point. The division of such numbers can also be done similarly to the "school" rule, but starting with the lower rather than higher digits of the number.
In various branches of mathematics, as well as in the application of mathematics in technology, there is often a situation where algebraic operations are performed not on numbers, but on objects of a different nature. For example, matrix addition, matrix multiplication, vector addition, operations on polynomials, operations on linear transformations, etc.
Definition 1. A ring is a set of mathematical objects in which two actions are defined - "addition" and "multiplication", which compare ordered pairs of elements with their "sum" and "product", which are elements of the same set. These actions meet the following requirements:
1.a+b=b+a(commutativity of addition).
2.(a+b)+c=a+(b+c)(associativity of addition).
3. There is a zero element 0 such that a+0=a, for any a.
4. For anyone a there is an opposite element − a such that a+(−a)=0.
5. (a+b)c=ac+bc(left distributivity).
5".c(a+b)=ca+cb(right distributivity).
Requirements 2, 3, 4 mean that the set of mathematical objects forms a group , and together with item 1 we are dealing with a commutative (Abelian) group with respect to addition.
As can be seen from the definition, in the general definition of a ring, no restrictions are imposed on multiplications, except for distributivity with addition. However, in various situations, it becomes necessary to consider rings with additional requirements.
6. (ab)c=a(bc)(associativity of multiplication).
7.ab=ba(commutativity of multiplication).
8. Existence of the identity element 1, i.e. such a 1=1 a=a, for any element a.
9. For any element of the element a there is an inverse element a−1 such that aa −1 =a −1 a= 1.
In various rings 6, 7, 8, 9 can be performed both separately and in various combinations.
A ring is called associative if condition 6 is satisfied, commutative if condition 7 is satisfied, commutative and associative if conditions 6 and 7 are satisfied. A ring is called a ring with a unit if condition 8 is satisfied.
Ring examples:
1. Set of square matrices.
Really. The fulfillment of points 1-5, 5 "is obvious. The zero element is the zero matrix. In addition, point 6 (associativity of multiplication), point 8 (the unit element is the identity matrix) are performed. Points 7 and 9 are not performed because in the general case, multiplication of square matrices is non-commutative, and also there is not always an inverse to a square matrix.
2. The set of all complex numbers.
3. The set of all real numbers.
4. The set of all rational numbers.
5. The set of all integers.
Definition 2. Any system of numbers containing the sum, difference and product of any two of its numbers is called number ring.
Examples 2-5 are number rings. Numeric rings are also all even numbers, as well as all integers divisible without remainder by some natural number n. Note that the set of odd numbers is not a ring since the sum of two odd numbers is an even number.
The natural numbers are not a ring, since 0 is not a natural number, and there are no natural opposites for natural numbers. The structure formed by natural numbers is called semicircle. More accurately,
semicircle is called a commutative semigroup with respect to addition and a semigroup with respect to multiplication, in which the operations of addition and multiplication are related by distributive laws.
We now introduce rigorous definitions of integers and prove their equivalence. Based on the concept of algebraic structures and the fact that the set of natural numbers is a semiring, but not a ring, we can introduce the following definition:
Definition 1. The ring of integers is the smallest ring containing the semiring of natural numbers.
This definition says nothing about the appearance of such numbers. In a school course, integers are defined as natural numbers, their opposites and 0. This definition can also be taken as the basis for constructing a strict definition.
Definition 2. A ring of integers is a ring whose elements are natural numbers, their opposites, and 0 (and only them).
Theorem 1. Definitions 1 and 2 are equivalent.
Proof: Denote by Z 1 the ring of integers in the sense of Definition 1, and by Z 2 the ring of integers in the sense of Definition 2. First we prove that Z 2 is included in Z 1 . Indeed, all elements of Z 2 are either natural numbers (they belong to Z 1, since Z 1 contains a semiring of natural numbers), or their opposites (they also belong to Z 1, since Z 1 is a ring, which means that for each element of this ring, there is an opposite one, and for each natural n н Z 1 , –n also belongs to Z 1), or 0 (0 н Z 1 , since Z 1 is a ring, and there is 0 in any ring), so any element from Z 2 also belongs to Z 1 , and hence Z 2 Í Z 1 . On the other hand, Z 2 contains a semiring of natural numbers, and Z 1 is the minimal ring containing natural numbers, that is, it cannot contain any another ring that satisfies this condition. But we have shown that it contains Z 2 , and therefore Z 1 = Z 2 . The theorem has been proven.
Definition 3. A ring of integers is a ring whose elements are all possible elements representable as a difference b - a (all possible solutions of the equation a + x = b), where a and b are arbitrary natural numbers.
Theorem 2. Definition 3 is equivalent to the two previous ones.
Proof: Denote by Z 3 the ring of integers in the sense of Definition 3, and by Z 1 = Z 2 , as before, the ring of integers in the sense of Definitions 1 and 2 (their equality has already been established). First we prove that Z 3 is included in Z 2 . Indeed, all elements of Z 3 can be represented as some differences of natural numbers b – a. For any two natural numbers, according to the trichotomy theorem, three options are possible:
In this case, the difference b – and is also a natural number and therefore belongs to Z 2 .
In this case, the difference of two equal elements will be denoted by the symbol 0. Let us prove that this is indeed the zero of the ring, that is, a neutral element with respect to addition. To do this, we use the definition of the difference a – a = x ó a = a + x and prove that b + x = b for any natural b. To prove it, it suffices to add the element b to the right and left sides of the equality a = a + x, and then use the reduction law (all these actions can be performed based on the known properties of rings). Zero belongs to Z 2 .
In this case, the difference a – b is a natural number, we denote
b - a \u003d - (a - b). We will prove that the elements a - b and b - a are indeed opposite, that is, they add up to zero. Indeed, if we denote a - b \u003d x, b - a \u003d y, then we get that a \u003d b + x, b \u003d y + a. Adding the equalities obtained term by term and reducing b, we get a \u003d x + y + a, that is, x + y \u003d a - a \u003d 0. Thus, a - b \u003d - (b - a) is a number opposite to the natural number, that is, it again belongs to Z2. Thus, Z 3 Н Z 2 .
On the other hand, Z 3 contains a semiring of natural numbers, since any natural number n can always be represented as
n = n / – 1 О Z 3 ,
and hence Z 1 Í Z 3 , since Z 1 is the minimal ring containing natural numbers. Using the already proven fact that Z 2 = Z 1 , we obtain Z 1 = Z 2 = Z 3 . The theorem has been proven.
Although at first glance it may seem that there are no axioms in the listed definitions of integers, these definitions are axiomatic, since all three definitions say that the set of integers is a ring. Therefore, the conditions from the definition of a ring serve as axioms in the axiomatic theory of integers.
Let's prove that the axiomatic theory of integers is consistent. To prove it, it is necessary to construct a model of the ring of integers using a known consistent theory (in our case, this can only be the axiomatic theory of natural numbers).
According to Definition 3, each integer can be represented as the difference of two natural numbers z = b – a. Associate with each integer z the corresponding pair
The introduced relation is reflexive, symmetric and transitive (the proof is left to the reader).
Like any equivalence relation, this relation generates a partition of the set of all possible pairs of natural numbers into equivalence classes, which we will denote as [
1) [
2) [
Let us show that the definitions introduced are correct, that is, they do not depend on the choice of specific representatives from the classes of pairs. In other words, if the pairs are equivalent
Proof: Apply the definition of pair equivalence:
Adding equalities (1) and (2) term by term, we get:
a + b 1 + c + d 1 \u003d b + a 1 + d + c 1.
All terms in the last equality are natural numbers, so we can apply the commutative and associative laws of addition, which leads us to the equality
(a + c) + (b 1 + d 1) \u003d (b + d) + (a 1 + c 1),
which is equivalent to the condition @ .
To prove the correctness of multiplication, we multiply equality (1) by c, we get:
ac + b 1 s \u003d bc + a 1 s.
Then we rewrite equality (1) as b + a 1 = a + b 1 and multiply by d:
bd + a 1 d = ad + b 1 d.
We add the resulting equalities term by term:
ac + bd + a 1 d + b 1 s = bc + ad + b 1 d + a 1 s,
which means that
Then we will do the same procedure with equality (2), only we will multiply it by a 1 and b 1. We get:
a 1 c + a 1 d 1 = a 1 d + a 1 c 1
b 1 d + b 1 c 1 \u003d b 1 c + b 1 d 1,
a 1 c + b 1 d + b 1 c 1 + a 1 d 1 = a 1 d + b 1 d + b 1 c 1 + a 1 c 1 ó
ó @
(here we have proved that ×
Thus, the correctness of the introduced definitions is proved.
Next, all properties of rings are directly verified: the associative law of addition and multiplication for classes of pairs, the commutative law of addition, and distributive laws. Let us give as an example the proof of the associative law of addition:
Since all components of pairs of numbers are natural
= <(a + c) +m), (b + d) +n)> =
= <(a + c), (b + d)> +
The remaining laws are verified in a similar way (note that a separate transformation of the left and right parts of the required equality to the same form can be a useful technique).
It is also necessary to prove the existence of a neutral element by addition. They can be a class of pairs of the form [<с, с>]. Really,
[
a + c + b = b + c + a (valid for any natural numbers).
In addition, for each class of pairs [
It can also be proved that the introduced set of pair classes is a commutative ring with a unit (the unit can be the class of pairs [
[
Let us check, using this rule, the validity of conditions C1 and C2 (from the definition of addition of natural numbers). Condition C1 (a + 1 = a /) in this case will be rewritten in the form:
[
[
a + c / +b = a + b + 1 + c = b + c + a +1 = b + c + a /
(Once again, we recall that all components are natural).
Condition C2 will look like:
[
We transform separately the left and right parts of this equality:
[
([
Thus, we see that the left and right sides are equal, which means that condition C2 is true. The proof of condition U1 is left to the reader. condition Y2 is a consequence of the distributive law.
So, the model of the ring of integers has been constructed, and, consequently, the axiomatic theory of integers is consistent if the axiomatic theory of natural numbers is consistent.
Properties of Operations on Integers:
2) a×(–b) = –a×b = –(ab)
3) – (– a) = a
4) (–a)×(–b) = ab
5) a×(–1) = – a
6) a - b \u003d - b + a \u003d - (b - a)
7) - a - b \u003d - (a + b)
8) (a - b) × c \u003d ac - bc
9) (a - b) - c \u003d a - (b + c)
10) a - (b - c) = a - b + c.
The proofs of all properties repeat the proofs of the corresponding properties for rings.
1) a + a × 0 = a × 1 + a × 0 = a × (1 + 0) = a × 1 = a, that is, a × 0 is a neutral element by addition.
2) a×(–b) + ab = a(–b + b) = a×0 = 0, i.e. the element a×(–b) is opposite to the element a×b.
3) (– a) + a = 0 (by definition of the opposite element). Similarly, (– a) + (– (– a)) = 0. Equating the left sides of the equalities and applying the law of reduction, we obtain – (– a) = a.
4) (–a)×(–b) = –(a×(–b)) = –(–(а×b)) = ab.
5) a×(–1) + a = a×(–1) + a×1 = a×(–1 + 1) = a×0 = 0
a×(–1) + a = 0
a×(–1) = –а.
6) By definition of the difference a - b, there is a number x such that a = x + b. Adding to the right and left sides of the equality -b on the left and using the commutative law, we obtain the first equality.
– b + a + b – a = –b + b + a – a = 0 + 0 = 0, which proves the second equality.
7) - a - b = - 1 × a - 1 × b = -1 × (a + b) = - (a + b).
8) (a – b) ×c = (a +(–1)× b) ×c = ac +(–1)×bc = ac – bc
9) (a - b) - c \u003d x,
a - b \u003d x + c,
a - (b + c) \u003d x, that is
(a - b) - c \u003d a - (b + c).
10) a - (b - c) = a + (- 1)×(b - c) = a + (- 1×b) + (-1)× (- c) = a - 1×b + 1× c = = a - b + c.
Tasks for independent solution
No. 2.1. In the right column of the table, find pairs equivalent to those given in the left column of the table.
| but)<7, 5> | 1) <5, 7> |
| b)<2, 3> | 2) <1, 10> |
| in)<10, 10> | 3) <5, 4> |
| G)<6, 2> | 4) <15, 5> |
| 5) <1, 5> | |
| 6) <9, 9> |
For each pair, indicate its opposite.
No. 2.2. Calculate
but) [<1, 5>] + [ <3, 2>]; b)[<3, 8>] + [<4, 7>];
in) [<7, 4>] – [<8, 3>]; G) [<1, 5>] – [ <3, 2>];
e) [<1, 5>] × [ <2, 2>]; e) [<2, 10>]× [<10, 2>].
No. 2.3. For the model of integers described in this section, check the commutative law of addition, the associative and commutative laws of multiplication, and distributive laws.
Examples
a + b i (\displaystyle a+bi) where a (\displaystyle a) And b (\displaystyle b) rational Numbers, i (\displaystyle i) is the imaginary unit. Such expressions can be added and multiplied according to the usual rules of operations with complex numbers, and each non-zero element has an inverse, as can be seen from the equality (a + bi) (aa 2 + b 2 − ba 2 + b 2 i) = (a + bi) (a − bi) a 2 + b 2 = 1. (\displaystyle (a+bi)\left(( \frac (a)(a^(2)+b^(2)))-(\frac (b)(a^(2)+b^(2)))i\right)=(\frac (( a+bi)(a-bi))(a^(2)+b^(2)))=1.) It follows from this that the rational Gaussian numbers form a field which is a two-dimensional space over (that is, a quadratic field).- More generally, for any square-free integer d (\displaystyle d) Q (d) (\displaystyle \mathbb (Q) ((\sqrt (d)))) will be a quadratic field expansion Q (\displaystyle \mathbb (Q) ).
- circular field Q (ζ n) (\displaystyle \mathbb (Q) (\zeta _(n))) obtained by adding Q (\displaystyle \mathbb (Q) ) primitive root n th power of unity. The field must also contain all its powers (that is, all roots n th power of unity), its dimension over Q (\displaystyle \mathbb (Q) ) equals the Euler function φ (n) (\displaystyle \varphi (n)).
- Real and complex numbers have infinite power over rational numbers, so they are not number fields. This follows from uncountability: any numeric field is countable.
- Field of all algebraic numbers A (\displaystyle \mathbb (A) ) is not numeric. Although the expansion A ⊃ Q (\displaystyle \mathbb (A) \supset \mathbb (Q) ) algebraically, it is not finite.
Ring of integers numeric field
Since the number field is an algebraic extension of the field Q (\displaystyle \mathbb (Q) ), any of its elements is a root of some polynomial with rational coefficients (that is, it is algebraic). Moreover, each element is a root of a polynomial with integer coefficients, since it is possible to multiply all rational coefficients by the product of the denominators. If a given element is a root of some unitary polynomial with integer coefficients, it is called an integer element (or an algebraic integer). Not all elements of a number field are integers: for example, it is easy to show that the only integer elements Q (\displaystyle \mathbb (Q) ) are regular integers.
It can be proved that the sum and product of two algebraic integers is again an algebraic integer, so the integer elements form a subring of the number field K (\displaystyle K) called whole ring fields K (\displaystyle K) and denoted by . The field does not contain zero divisors and this property is inherited when passing to a subring, so the ring of integers is integral; private ring box O K (\displaystyle (\mathcal (O))_(K)) is the field itself K (\displaystyle K). The ring of integers of any number field has the following three properties: it is integrally closed, Noetherian, and one-dimensional. A commutative ring with these properties is called Dedekind, after Richard Dedekind.
Decomposition into primes and a group of classes
In an arbitrary Dedekind ring, there is a unique decomposition of non-zero ideals into a product of simple ones. However, not every ring of integers satisfies the factorial property: already for the ring of integers, a quadratic field OQ (− 5) = Z [ − 5 ] (\displaystyle (\mathcal (O))_(\mathbb (Q) ((\sqrt (-5))))=\mathbb (Z) [(\sqrt ( -five))]) decomposition is not unique:
6 = 2 ⋅ 3 = (1 + − 5) (1 − − 5) (\displaystyle 6=2\cdot 3=(1+(\sqrt (-5)))(1-(\sqrt (-5) )))By introducing a norm on this ring, we can show that these expansions are indeed different, that is, one cannot be obtained from the other by multiplying by an invertible element.
The degree of violation of the factorial property is measured using the ideal class group, this group for the ring of integers is always finite and its order is called the number of classes.
Number field bases
whole basis
whole basis number field F degree n- it's a set
B = {b 1 , …, b n}from n elements of the ring of integer fields F, such that any element of the ring of integers O F fields F can only be written as Z-linear combination of elements B; that is, for any x from O F there is a unique decomposition
x = m 1 b 1 + … + m n b n,where m i are regular integers. In this case, any element F can be written as
m 1 b 1 + … + m n b n,where m i are rational numbers. After this the whole elements F are distinguished by the property that these are exactly those elements for which all m i whole.
Using tools such as localization and the Frobenius endomorphism, one can construct such a basis for any number field. Its construction is a built-in feature in many computer algebra systems.
Power basis
Let be F- numeric degree field n. Among all possible bases F(how Q-vector space), there are power bases, that is, bases of the form
B x = {1, x, x 2 , …, x n−1 }for some x ∈ F. According to the primitive element theorem, such x always exists, it is called primitive element this extension.
Norm and trace
An algebraic number field is a finite-dimensional vector space over Q (\displaystyle \mathbb (Q) )(we denote its dimension as n (\displaystyle n)), and multiplication by an arbitrary element of the field is a linear transformation of this space. Let be e 1 , e 2 , … e n (\displaystyle e_(1),e_(2),\ldots e_(n))- any basis F, then the transformation x ↦ α x (\displaystyle x\mapsto \alpha x) corresponds matrix A = (a i j) (\displaystyle A=(a_(ij))), determined by the condition
α e i = ∑ j = 1 n a i j e j , a i j ∈ Q . (\displaystyle \alpha e_(i)=\sum _(j=1)^(n)a_(ij)e_(j),\quad a_(ij)\in \mathbf (Q) .)
The elements of this matrix depend on the choice of the basis, however, all matrix invariants, such as determinant and trace, do not depend on it. In the context of algebraic extensions, the determinant of an element multiplication matrix is called the norm this element (denoted N (x) (\displaystyle N(x))); matrix trace - trace element(denoted Tr (x) (\displaystyle (\text(Tr))(x))).
The trace of an element is a linear functional on F:
Tr (x + y) = Tr (x) + Tr (y) (\displaystyle (\text(Tr))(x+y)=(\text(Tr))(x)+(\text(Tr)) (y)) And Tr (λ x) = λ Tr (x) , λ ∈ Q (\displaystyle (\text(Tr))(\lambda x)=\lambda (\text(Tr))(x),\lambda \in \mathbb (Q) ).The norm is a multiplicative and homogeneous function:
N (x y) = N (x) ⋅ N (y) (\displaystyle N(xy)=N(x)\cdot N(y)) And N (λ x) = λ n N (x) , λ ∈ Q (\displaystyle N(\lambda x)=\lambda ^(n)N(x),\lambda \in \mathbb (Q) ).As the initial basis, you can choose an integer basis, multiplication by an integer algebraic number (that is, by an element of the ring of integers) in this basis will correspond to a matrix with integer elements. Therefore, the trace and norm of any element of the ring of integers are integers.
An example of using a norm
Let be d (\displaystyle d)- - an integer element, since it is the root of the reduced polynomial x 2 − d (\displaystyle x^(2)-d)). In this basis, multiplication by a + b d (\displaystyle a+b(\sqrt (d))) corresponds matrix
(a d b b a) (\displaystyle (\begin(pmatrix)a&db\\b&a\end(pmatrix)))Consequently, N (a + b d) = a 2 − d b 2 (\displaystyle N(a+b(\sqrt (d)))=a^(2)-db^(2)). On the elements of the ring, this norm takes integer values. The norm is a homomorphism of the multiplicative group Z [ d ] (\displaystyle \mathbb (Z) [(\sqrt (d))]) per multiplicative group Z (\displaystyle \mathbb (Z) ), so the norm of invertible elements of a ring can only be equal to 1 (\displaystyle 1) or − 1 (\displaystyle -1). To solve Pell's equation a 2 − d b 2 = 1 (\displaystyle a^(2)-db^(2)=1), it suffices to find all invertible elements of the ring of integers (also called ring units) and select among them those with the norm 1 (\displaystyle 1). According to Dirichlet's unit theorem, all invertible elements of a given ring are powers of one element (up to multiplication by − 1 (\displaystyle -1)), therefore, to find all solutions of the Pell equation, it suffices to find one fundamental solution.
see also
Literature
- H. Koch. Algebraic Number Theory. - M.: VINITI, 1990. - T. 62. - 301 p. - (Results of science and technology. Series "Modern problems of mathematics. Fundamental directions".).
- Chebotarev N.G. Fundamentals of Galois theory. Part 2. - M.: Editorial URSS, 2004.
- Weil G. Algebraic number theory. Per. from English. - M. : Editorial URSS, 2011.
- Serge Lang, Algebraic Number Theory, second edition, Springer, 2000
