In various branches of mathematics, as well as in the application of mathematics in technology, a situation often occurs when 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 associate ordered pairs of elements with their “sum” and “product”, which are elements of the same set. These actions satisfy 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 point 1 we are dealing with a commutative (Abelian) group with respect to addition.

As can be seen from the definition, in general definition rings, no restrictions are imposed on multiplications other than distributivity with addition. However, in different 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. The existence of a single element 1, i.e. such a·1=1· a=a, for any element a.

9. For any item 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 either separately or 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 identity if condition 8 is satisfied.

Examples of rings:

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 is fulfilled (the unit element is the unit matrix). Points 7 and 9 are not fulfilled because in the general case, multiplication of square matrices is non-commutative, and also the inverse of a square matrix does not always exist.

2. The set of all complex numbers.

3. Lots of everyone 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. Number rings are also all even numbers, as well as all integers divisible without a remainder by some natural number n. Note that the set of odd numbers is not a ring because the sum of two odd numbers is an even number.

From a programming course we know that an integer can be represented in computer memory in different ways, in particular, this representation depends on how it is described: as a value of type integer, or real, or string. Moreover, in most programming languages, integers are understood to be numbers from a very limited range: a typical case is from -2 15 = -32768 to 2 15 - 1 = 32767. Systems computer algebra deal with large integers, in particular, any such system can calculate and display numbers of the form 1000 in decimal notation! (more than a thousand characters).

In this course, we will consider the representation of integers in symbolic form and will not go into detail about what memory is allocated for recording one character (bit, byte, or other). The most common is to represent integers in positional number systems. Such a system is determined by the choice of number base, for example, 10. The set of decimal integers is usually described as follows:

The written definition of integers gives an unambiguous representation of each such number, and a similar definition (only, perhaps, with a different basis) is used in most systems computer algebra. Using this representation, it is convenient to implement arithmetic operations on integers. Moreover, addition and subtraction are relatively “cheap” operations, while multiplication and division are “expensive”. When assessing the complexity of arithmetic operations, one should take into account both the cost of an elementary operation (single-digit) and the number of single-digit operations to perform any operation on multi-digit numbers. The complexity of multiplication and division is due, first of all, to the fact that as the length of a number increases (its recording in any number system), the number of elementary operations increases according to a quadratic law, in contrast to the linear law for addition and subtraction. In addition, what we usually call an algorithm for dividing multi-digit numbers is actually based on a search (often quite significant) of the possible next digit of the quotient, and it is not enough to simply use the rules for dividing single-digit numbers. If the base of the number system is large (often it can be on the order of 2 30), this method is ineffective.

Let be a natural number (written in the decimal system). To get his record in the -ary number system, you can use the following algorithm (denotes the integer part of the number):

Given: A-natural number in the decimal number system k > 1-natural number Needed: A-recording of the number A in the k-ary number system Start i:= 0 cycle until A > 0 bi:= A (mod k) A:= i:= i + 1 end of cycle dA:= i - 1 end

To restore a decimal number from the sequence of its k-ary notation, the following algorithm is used:

Given: k > 1-natural number sequence of digits representing the number A in the k-ary system Needed: A-recording of the number A in the decimal number system Start A:= 0 cycle until the end of the sequence b:= next element of the sequence A:= A * k + b end of loop End

1.2. EXERCISE. Explain why division is used to convert a number from the decimal system to the k-ary system, and multiplication is used to convert from the k-ary system to the decimal system.

By multiplying two two-digit numbers in the decimal number system with a “column”, we perform the following operations:

(10a + b)(10c + d) = 100ac + 10(ad + bc) + bd,

i.e. 4 operations of multiplication of single-digit numbers, 3 operations of addition and 2 operations of multiplication by a power of the radix, which are reduced to a shift. When assessing complexity, you can take into account all elementary operations without dividing them by weights (in this example we have 9 elementary operations). With this approach, the problem of optimizing the algorithm is reduced to minimizing the total number of elementary operations. It can, however, be considered that multiplication is a more “expensive” operation than addition, which, in turn, is “more expensive” than shift. Considering only the most expensive operations, we get that multiplicative The difficulty of multiplying two-digit numbers in a column is 4.

Section 5 discusses algorithms for calculating greatest common divisors and evaluates their complexity.

The representation considered is not the only canonical representation of integers. As already noted, to choose a canonical representation, you can use the uniqueness of the decomposition of a natural number into prime factors. This representation of an integer can be used in those tasks where only multiplication and division operations are used, since they become very “cheap”, but the cost of addition and subtraction operations increases disproportionately, which prevents the use of such a representation. In some problems, abandoning the canonical representation gives a significant gain in performance; in particular, partial factorization of a number can be used. A similar method is especially useful when working not with numbers, but with polynomials.

If it is known that during the operation of the program, all integers encountered in calculations are limited in absolute value by some given constant, then to define such numbers, one can use their system of residues modulo some coprime numbers, the product of which exceeds the mentioned constant. Calculations with residue classes are generally faster than multiple-precision arithmetic. And with this approach, multiple precision arithmetic should be used only when entering or outputting information.

Note that, along with canonical representations in systems computer algebra Other representations are also used. In particular, it is desirable that the presence or absence of a “+” sign in front of an integer does not affect the computer’s perception of it. Thus, for positive numbers, an ambiguous representation is obtained, although the form of negative numbers is uniquely determined.

Another requirement is that the perception of a number should not be affected by the presence of zeros before the first significant digit.

1.3. EXERCISES.

1. Estimate the number of single-digit multiplications used when multiplying an m-digit number by a n-digit column.
2. Show that two two-digit numbers can be multiplied using only 3 single-digit multiplications and increasing the number of additions.
3. Find an algorithm for dividing long numbers that does not require a lot of searching when finding the first digit of the quotient.
4. Describe the translation algorithm natural numbers from m-ary number system to n-ary number system.
5. IN roman numbering The following symbols are used to write numbers: I - one, V - five, X - ten, L - fifty, C - one hundred, D - five hundred, M - thousand. A symbol is considered negative if there is a symbol of a larger number to the right of it, and positive otherwise. For example, the number 1948 in this system will be written like this: MCMXLVIII. Formulate an algorithm for converting a number from Roman to decimal and back. Implement the resulting algorithm in one of the algorithmic languages ​​(for example, C). Limitations on source data: 1<= N < 3700 , в записи результата ни один символ не должен появляться больше 3 раз.
6. Formulate an algorithm and write a program for adding natural numbers in Roman numeration.
7. We will say that we are dealing with a number system with mixed or vector basis, if we are given a vector of n natural numbers M = (m 1 , . . . , m n) (radix) and the notation K = (k 0 , k 1 , . . . , k n) denotes the number k = k 0 +m 1 (k 1 +m 2 (k 2 +· · ·+m n ·k n)...)). Write a program that, based on data (day of the week, hours, minutes, seconds), determines how many seconds have passed since the beginning of the week (Monday, 0, 0, 0) = 0, and performs the reverse transformation.

Examples

a + b i (\displaystyle a+bi) Where a (\displaystyle a) And b (\displaystyle b) rational numbers, i (\displaystyle i)- imaginary unit. Such expressions can be added and multiplied according to the usual rules for 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 extension Q (\displaystyle \mathbb (Q) ).
• Circular field Q (ζ n) (\displaystyle \mathbb (Q) (\zeta _(n))) obtained by adding to Q (\displaystyle \mathbb (Q) ) primitive root n-th power of unity. The field must contain all its powers (that is, all the roots n th power of unity), its dimension is over Q (\displaystyle \mathbb (Q) ) equals Euler's function φ (n) (\displaystyle \varphi (n)).
• Real and complex numbers have infinite powers over rational numbers, so they are not number fields. This follows from uncountability: any number 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) ) algebraic, it is not finite.

Number field integer ring

Since the number field is an algebraic extension of the field Q (\displaystyle \mathbb (Q) ), any of its elements is the 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 all rational coefficients can be multiplied by the product of the denominators. If this element is the 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 elements are integers Q (\displaystyle \mathbb (Q) ) are ordinary integers.

It can be proven 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 a ring of whole 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 field O K (\displaystyle (\mathcal (O))_(K))- this 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 such properties is called a Dedekind ring, after Richard Dedekind.

Primer decomposition and class group

In an arbitrary Dedekind ring, there is a unique decomposition of nonzero ideals into a product of primes. However, not every ring of integers satisfies the factoriality property: already for the ring of integers of a quadratic field O Q (− 5) = Z [ − 5 ] (\displaystyle (\mathcal (O))_(\mathbb (Q) ((\sqrt (-5))))=\mathbb (Z) [(\sqrt ( -5))]) the 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 factoriality property is measured using the group of ideal classes; this group for a 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- this is a lot

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 The only way to write it is as Z-linear combination of elements B; that is, for anyone x from O F there is only one decomposition

x = m 1 b 1 + … + m n b n,

Where m i- ordinary integers. In this case, any element F can be written as

m 1 b 1 + … + m n b n,

Where m i- rational numbers. After this are whole elements F are distinguished by the property that these are exactly those elements for which everything m i whole.

Using tools such as localization and 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- degree numeric field n. Among all possible bases F(How Q-vector space), there are power bases, that is, bases of the form

Bx = {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 field element is a linear transformation of this space. Let e 1 , e 2 , … e n (\displaystyle e_(1),e_(2),\ldots e_(n))- some basis F, then the transformation x ↦ α x (\displaystyle x\mapsto \alpha x) corresponds to 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 basis, but all invariants of the matrix, such as the 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 - next 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) ).

You can choose an integer basis as the initial 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.

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 to 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 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) ), therefore the norm of invertible elements of the 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 is enough to find all invertible elements of the ring of integers (also called ring units) and identify among them those that have the norm 1 (\displaystyle 1). According to Dirichlet's unity 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 to the Pell equation, it is enough 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. Basics 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

A ring in which the relation “to be greater than zero” (denoted a > 0) is introduced is called located ring, if for any elements of this ring two conditions are met:

1) one and only one of the conditions is true

a > 0 \/ –a >0 \/ a = 0

2) a > 0 /\ b > 0 => a + b > 0 /\ ab > 0.

A set in which some order relation is introduced - non-strict (reflexive, antisymmetric and transitive) or strict (anti-reflexive and transitive) is called ordered. If the law of trichotomy is satisfied, then the set is called linear ordered. If we consider not an arbitrary set, but some algebraic system, for example, a ring or a field, then for the ordering of such a system we also introduce monotonicity requirements regarding the operations introduced in this system (algebraic structure). So ordered ring/field is a non-zero ring/field in which a linear order relation (a > b) is introduced that satisfies two conditions:

1) a > b => a + c > b + c;

2) a > b, c > 0 => a c > b c;

Theorem 1. Any located ring is an ordered system (ring).

Indeed, if the relation “to be greater than 0” is introduced in a ring, then the relation greater than for two arbitrary elements can be introduced if we assume that

a > b  a – b > 0.

This attitude is a strict attitude, linear order.

This “greater than” relation is anti-reflexive, since the condition a > a is equivalent to the condition a – a > 0, the latter contradicts the fact that a – a = 0 (according to the first condition of the located ring, an element cannot be both greater than 0 and equal to 0) . Thus, the statement a > a is false for any element a, so the relation is anti-reflexive.

Let's prove transitivity: if a > b and b > c, then a > c. By definition, from the conditions of the theorem it follows that a – b > 0 and b – c > 0. By adding these two elements greater than zero, we again obtain an element greater than zero (according to the second condition of the located ring):

a – b + b – c = a – c > 0.

The latter means that a > c. Thus, the introduced relation is a relation of strict order. Moreover, this relation is a relation of linear order, that is, for the set of natural numbers, trichotomy theorem:

For any two natural numbers, one and only one of the following three statements is true:

Indeed (due to the first condition of the located ring) for the number a – b one and only one of the conditions is true:

1) a – b > 0 = > a > b

2) – (a – b) = b – a > 0 => b > a

3) a – b = 0 = > a = b.

The monotonicity properties also hold for any located ring. Really

1) a > b => a – b > 0 = > a + c – c – b > 0 = > a + c > b + c;

2) a > b /\ c > 0 => a – b > 0 => (according to the second condition of the located ring) (a – b)c > 0 => ac – bc > 0 => ac > bc.

Thus, we have proven that any arranged ring is an ordered ring (an ordered system).

For any located ring the following properties will also be valid:

a) a + c > b + c => a > b;

b) a > b /\ c > d => a + c > b + d;

c) a > b /\ c< 0=>ac< bc;

The same properties occur for other signs<, , .

Let us prove, for example, property (c). By definition, from the condition a > b it follows that a – b > 0, and from the condition c< 0 (0 >c) it follows that 0 – c > 0, and therefore the number – c > 0, multiply two positive numbers (a – b)(–c). The result will also be positive according to the second condition of the located ring, that is

(a – b)(–c) > 0 => –ac + bc > 0 => bc – ac > 0 => bc > ac => ac< bc,

Q.E.D.

d) aa = a 2  0;

Proof: According to the first condition of the located ring, either a > 0, or –a > 0, or a = 0. Let us consider these cases separately:

1) a > 0 => aa > 0 (according to the second condition of the located ring) => a 2 > 0.

2) –а > 0 => (–а)(–а) > 0, but by the property of the ring (–а)(–а) = аа = a 2 > 0.

3) a = 0 => aa = a 2 = 0.

Thus, in all three cases a 2 is either greater than zero or equal to 0, which just means that a 2 ≥ 0 and the property is proven (note that we also proved that the square of an element of a located ring is 0 if and only if the element itself is 0).

e) ab = 0  a = 0 \/ b = 0.

Proof: Assume the opposite (ab =0, but neither a nor b are equal to zero). Then only two options are possible for a, either a > 0, or a > 0 (the option a = 0 is excluded by our assumption). Each of these two cases splits into two more cases depending on b (either b > 0 or – b > 0). Then there are 4 options:

a > 0, b > 0 => ab > 0;

– a > 0, b > 0 => ab< 0;

a >0, – b > 0 => ab< 0;

– a > 0 –b > 0 => ab > 0.

As we see, each of these cases contradicts the condition ab = 0. The property has been proven.

The last property means that the located ring is an area of ​​integrity, which is also a mandatory property of ordered systems.

Theorem 1 shows that any located ring is an ordered system. The converse is also true - any ordered ring is located. Indeed, if a ring has a relation a > b and any two elements of the ring are comparable to each other, then 0 is comparable to any element a, that is, either a > 0 or a< 0, либо а = 0, что почти точно совпадает с первым условием расположенного кольца, требуется только доказать, что условие а < 0 равносильно в упорядоченном кольце условию –a >0. In order to prove the latter, we apply the property of monotonicity of ordered systems: to the right and left sides of the inequality a< 0 прибавим –а, в результате чего получим требуемое.

The second condition for a located ring follows from the properties of monotonicity and transitivity:

a > 0, b > 0 => a + b > 0 + b = b > 0 => a +b >0,

a > 0, b > 0 => ab > 0b = 0 => ab > 0.

Theorem 2. The ring of integers is a arranged ring (an ordered system).

Proof: Let's use definition 2 of the ring of integers (see 2.1). According to this definition, any integer is either a natural number (the number n is given as [ ], or the opposite of natural (– n corresponds to the class [<1, n / >] , or 0 (class [<1, 1>]). Let us introduce the definition of “to be greater than zero” for integers according to the rule:

a > 0  a  N

Then the first condition of the located ring is automatically satisfied for integers: if a is a natural number, then it is greater than 0, if a is the opposite of a natural number, then –a is a natural number, that is, also greater than 0, the option a = 0 is also possible, which also makes true disjunction in the first condition of the located ring. The validity of the second condition of the located ring follows from the fact that the sum and product of two natural numbers (integers greater than zero) is again a natural number, and therefore greater than zero.

Thus, all properties of the located rings are automatically transferred to all integers. In addition, the discreteness theorem holds for integers (but not for arbitrary arranged rings):

Discreteness theorem. You cannot insert any integer between two adjacent integers:

( a, x  Z) .

Proof: we will consider all possible cases for a, and we will assume the opposite, that is, that there exists an x ​​such that

A< x < a +1.

1) if a is a natural number, then a + 1 is a natural number. Then, according to the discreteness theorem for natural numbers, no natural number x can be inserted between a and a / = a + 1, that is, x, in any case, cannot be a natural number. If we assume that x = 0, then our assumption is that

A< x < a +1

will lead us to condition a< 0 < a + 1 (здесь мы просто подставили х = 0), откуда видим, что а < 0, что противоречит тому, что а – натуральное. Если х не натуральное и не 0, но х – целое, значит –х – натуральное, тогда х < 0. При этом, согласно нашему предположению, а < x, что по свойству транзитивности опять приводит к тому, что а < 0, что невозможно, так как а – натуральное. Таким образом, для натуральных а утверждение а < x < a +1 всегда ложно, и теорема справедлива.

2) a = 0. Then a + 1 = 1. If condition a is satisfied< x < a + 1, то мы получим 0 < x < 1, то есть с одной стороны х – натуральное (целое, большее нуля), а с другой стороны х меньше 1, что невозможно для натуральных чисел. Таким образом, для нуля наша теорема справедлива.

3) a is negative (–a > 0), then a + 1  0. If a + 1< 0, то умножая условие а < x < a + 1 на –1 получим:

–a–1< – x < –a,

that is, we come to the situation considered in the first case (since both –a–1 and –a are natural), from which – x cannot be an integer, and therefore x cannot be an integer. The situation when a + 1 = 0 means that a = –1, that is

–1 < x < 0.

By multiplying this inequality by (–1), we arrive at case 2. Thus, the theorem is valid in all situations.

Tower of Archimedes. For any integer a and integer b > 0, there exists a natural number n such that a< bn.

For a natural a, the theorem has already been proven, since the condition b > 0 means that the number b is natural. For a  0 the theorem is also obvious, since the right side bn is a natural number, that is, also greater than zero.

In a ring of integers (as in any arranged ring), we can introduce the concept of a modulus:

|a| = .

The properties of the modules are fair:

1) |a + b|  |a| + |b|;

2) |a – b|  |a| – |b|;

3) |a  b| = |a|  |b|.

Proof: 1) Note that from the definition it is obvious that |a| is a quantity that is always non-negative (in the first case |a| = a ≥ 0, in the second |a| = –a, but a< 0, откуда –а >0). The inequalities |a| ≥ a, |a| ≥ –a (the modulus is equal to the corresponding expression if it is non-negative, and greater than it if it is negative). Similar inequalities are valid for b: |b| ≥ b, |b| ≥ –b. Adding the corresponding inequalities and applying property (b) of arranged rings, we obtain

|a| + |b| ≥ a + b |a| + |b| ≥ – a – b.

According to the module definition

|a+b| =
,

but both expressions on the right side of the equality, as shown above, do not exceed |a| + |b|, which proves the first property of modules.

2) Replace a in the first property with a – b. We get:

|a – b + b| ≤ |a – b| + |b|

|a | ≤ |a – b| + |b|

Let's move |b| from the right side to the left with the opposite sign

|a| – | b| ≤ |a – b| =>|a – b|  |a| – |b|.

The proof of Property 3 is left to the reader.

Task: Solve the equation in whole numbers

2y 2 + 3xy – 2x 2 + x – 2y = 5.

Solution: Let's factorize the left side. To do this, imagine the term 3xy = – xy + 4xy

2y 2 + 3xy – 2x 2 + x – 2y = 2y 2 – xy + 4xy – 2x 2 + x – 2y =

Y(2y – x) + 2x(2y – x) – (2y – x) = (y + 2x – 1)(2y – x).

Thus, our equation can be rewritten as

(y + 2x – 1)(2y – x) = 5.

Since we need to solve it in integers, x and y must be integers, which means that the factors on the left side of our equation are also integers. The number 5 on the right side of our equation can be represented as a product of integer factors in only 4 ways:

5 = 51 = 15 = –5(–1) = –1(–5). Therefore, the following options are possible:

1)
2)
3)
4)

Among the listed systems, only (4) has an integer solution:

x = 1, y = –2.

Tasks for independent solution

No. 2.4. For elements a, b, c, d of an arbitrary located ring, prove the properties:

a) a + c > b + c => a > b; b) a > b /\ c > d => a + c > b + d.

No. 2.5. Solve the equations in whole numbers:

a) y 2 – 2xy – 2x = 6; b) 2x 2 – 11xy + 12y 2 = 17; c) 35xy + 5x – 7y = 1; d) x 2 – 3xy + 2y 2 = 3; d) ;

e) xy + 3x – 5y + 3 = 0; g) 2xy – 3y 2 – 4y + 2x = 2; h) xy 2 + x = 48; i) 1! + 2! + 3! + … + n! = y 2 ; j) x 3 – 2y 3 – 4z 3 = 0

No. 2.6. Find a four-digit number that is a perfect square and such that its first two digits are equal and its last two digits are equal.

No. 2.7. Find a two-digit number equal to the sum of its tens and the square of its units.

No. 2.8. Find a two-digit number that is equal to twice the product of its digits.

No. 2.9. Prove that the difference between a three-digit number and a number written with the same digits in reverse order cannot be the square of a natural number.

No. 2.10. Find all natural numbers ending in 91, which, after crossing out these digits, are reduced by an integer factor.

No. 2.11. Find a two-digit number equal to the square of its units added to the cube of its tens.

No. 2.12. Find a six-digit number starting with the digit 2, which increases by 3 times when this digit is moved to the end of the number.

No. 2.13. There are more than 40 but less than 48 integers written on the board. The arithmetic mean of all these numbers is – 3, the arithmetic mean of the positive ones is 4, and the arithmetic mean of the negative ones is – 8. How many numbers are written on the board? Which numbers are greater, positive or negative? What is the maximum possible number of positive numbers?

No. 2.14. Can the quotient of a three-digit number and the sum of its digits be equal to 89? Could this quotient be equal to 86? What is the maximum possible value of this quotient?

Federal Agency for Education

State educational institution higher professional education

Vyatka State Humanitarian University

Faculty of Mathematics

Department of Mathematical Analysis and Methods
teaching mathematics

Final qualifying work

on the topic: Gaussian integer ring.

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 at the State Attestation Commission

Head Department________________ Vechtomov E.M.

« »________________

Dean of the Faculty __________________ Varankina V.I.

Introduction.

Ring of complex integers

was discovered by Carl Gauss and named Gaussian in his honor.

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 a given 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 in the ring of complex numbers there are only four invertible elements: ; proved the validity of the theorem on division with remainder, the theorem on the uniqueness of factorization; showed which prime natural numbers will remain prime in a ring; discovered the nature of simple integers and complex numbers.

The theory developed by K. Gauss, described in his work Arithmetic Studies, was a fundamental discovery for the theory of numbers and algebra.

The following goals were set in the final work:

1. Develop the theory of divisibility in the ring of Gaussian numbers.

2. Find out the nature of prime Gaussian numbers.

3. Show the use of Gaussian numbers in solving ordinary Diophantine problems.

CHAPTER 1. DIVISION IN THE RING OF GAUSS NUMBERS.

Let's consider the set of complex numbers. By analogy with the set of real numbers, a certain subset of integers can be distinguished in it. Set of numbers of the form

, Where we call them complex integers or Gaussian numbers. It is easy to verify that the ring axioms hold for this set. Thus, this set of complex numbers is a ring and is called ring of Gaussian integers . Let us denote it as , since it is an extension of the ring by element: .

Since the ring of Gaussian numbers is a subset of complex numbers, some definitions and properties of complex numbers are valid for it. So, for example, for each Gaussian number

corresponds to a vector with a beginning at a point and an end at . Hence, module there are Gaussian numbers. Note that in the set under consideration, the submodular 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. Thus . The following properties of the norm can be distinguished. 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 verified 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 contractility of the ring

, that is (6)

1.1 REVERSIBLE AND ALLIED ELEMENTS.

Let's see which Gaussian numbers will be invertible. Multiplication neutral is

. If a Gaussian number reversible , then, by definition, there exists such that . Moving on to the norms, according to property 3, we obtain . But these norms are natural, therefore. This means, by property 4, . Conversely, all elements of this set are invertible, since . Consequently, numbers with norm equal to one will be invertible, that is, , .

As you can see, not all Gaussian numbers will be invertible. It is therefore interesting to consider the issue of divisibility. As usual, we say that

shares on , if there is such that . For any Gaussian numbers, as well as invertible ones, the properties are valid. (7) (8) (9) (10) , where (11) (12)

Easily checked (8), (9), (11), (12). Justice (7) follows from (2), and (10) follows from (6). Due to property (9), the elements of the set