We have seen that operations on polynomials are reduced to operations on their coefficients. At the same time, for addition, subtraction and multiplication of polynomials, three arithmetic operations are sufficient - division of numbers was not required. Since the sum, difference, and product of two real numbers are again real numbers, adding, subtracting, and multiplying polynomials with real coefficients results in polynomials with real coefficients.

However, one does not always have to deal with polynomials that have any real coefficients. There are cases when, by the very essence of the matter, the coefficients should have only integer or only rational values. Depending on what values ​​of the coefficients are considered admissible, the properties of the polynomials change. For example, if we consider polynomials with any real coefficients, then we can factorize:

If we confine ourselves to polynomials with integer coefficients, then expansion (1) does not make sense and we must consider the polynomial to be indecomposable into factors.

This shows that the theory of polynomials essentially depends on what coefficients are considered admissible. Far from any set of coefficients can be taken as acceptable. For example, consider all polynomials whose coefficients are odd integers. It is clear that the sum of two such polynomials will no longer be a polynomial of the same type: after all, the sum of odd numbers is an even number.

Let us pose the question: what are “good” sets of coefficients? When does the sum, difference, product of polynomials with coefficients of a given type have coefficients of the same type? To answer this question, we introduce the notion of a number ring.

Definition. A non-empty set of numbers is called a number ring if, together with any two numbers a and , it contains their sum, difference, and product. This is also expressed more briefly by saying that the number ring is closed under the operations of addition, subtraction and multiplication.

1) The set of integers is a numerical ring: the sum, difference and product of integers are integers. The set of natural numbers is not a numerical ring, since the difference of natural numbers can be negative.

2) The set of all rational numbers is a numerical ring, since the sum, difference and product of rational numbers are rational.

3) Forms a number ring and the set of all real numbers.

4) Numbers of the form a where a and integers form a numerical ring. This follows from the relations:

5) The set of odd numbers is not a number ring, since the sum of odd numbers is even. The set of even numbers is a numerical ring.

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 + b i) (a a 2 + b 2 − b a 2 + b 2 i) = (a + b i) (a − b i) 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 O Q (− 5) = Z [ − 5 ] (\displaystyle (\mathcal (O))_(\mathbb (Q) ((\sqrt (-5))))=\mathbb (Z) [(\sqrt ( -5))]) 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 degrees 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 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 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 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)))

Hence, 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 having 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

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 unit ring 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.

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 define 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 therefore, 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 p-adic integers 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-adic 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.