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The ring of rational numbers is a field. The concept of a ring, the simplest properties of rings
Definition 2.5. Ring called algebra
R = (R, +, ⋅, 0 , 1 ),
whose signature consists of two binary and two nullary operations, and for any a, b, c ∈ R the following equalities hold:
- a+(b+c) = (a+b)+c;
- a+b = b+a;
- a + 0 = a;
- for every a ∈ R there is an element a" such that a+a" = 0
- a-(b-c) = (a-b)-c;
- a ⋅ 1 = 1 ⋅ a = a;
- a⋅(b + c) = a⋅b + a⋅c, (b + c) ⋅ a = b⋅ a + c⋅a.
The operation + is called adding the ring , operation ring multiplication , element 0 - zero of the ring , element 1 - ring unit .
Equalities 1-7 specified in the definition are called axioms of the ring . Let us consider these equalities from the point of view of the concept groups And monoid.
Ring axioms 1-4 mean that the algebra (R, +, 0 ), the signature of which consists only of the operations of addition of the ring + and zero of the ring 0 , is abelian group. This group is called additive group of the ring R and they also say that by addition a ring is a commutative (Abelian) group.
Ring axioms 5 and 6 show that the algebra (R, ⋅, 1), whose signature includes only the multiplication of the ring ⋅ and the identity of the ring 1, is a monoid. This monoid is called multiplicative monoid of the ring R and they say that by multiplication a ring is a monoid.
The connection between addition of a ring and multiplication of a ring is established by Axiom 7, according to which the operation of multiplication is distributive with respect to the operation of addition.
Considering the above, we note that a ring is an algebra with two binary and two nullary operations R =(R, +, ⋅, 0 , 1 ), such that:
- algebra (R, +, 0 ) - commutative group;
- algebra (R, ⋅, 1 ) - monoid;
- the operation ⋅ (multiplication of a ring) is distributive with respect to the operation + (addition of a ring).
Remark 2.2. In the literature there is a different composition of ring axioms related to multiplication. Thus, axiom 6 may be absent (there is no 1 ) and axiom 5 (multiplication is not associative). In this case, associative rings are distinguished (the requirement of associative multiplication is added to the axioms of the ring) and rings with unity. In the latter case, the requirements of associativity of multiplication and the existence of a unit are added.
Definition 2.6. The ring is called commutative , if its multiplication operation is commutative.
Example 2.12. A. The algebra (ℤ, +, ⋅, 0, 1) is a commutative ring. Note that the algebra (ℕ 0, +, ⋅, 0, 1) will not be a ring, since (ℕ 0, +) is a commutative monoid, but not a group.
b. Consider the algebra ℤ k = ((0,1,..., k - 1), ⊕ k , ⨀ k , 0,1) (k>1) with the operation ⊕ k of addition modulo l and ⨀ k (multiplication modulo l). The latter is similar to the operation of addition modulo l: m ⨀ k n is equal to the remainder of division by k of the number m ⋅ n. This algebra is a commutative ring, which is called ring of residues modulo k.
V. The algebra (2 A, Δ, ∩, ∅, A) is a commutative ring, which follows from the properties of intersection and symmetric difference of sets.
G. An example of a non-commutative ring gives the set of all square matrices of a fixed order with the operations of matrix addition and multiplication. The unit of this ring is the identity matrix, and the zero is the zero matrix.
d. Let L- linear space. Let us consider the set of all linear operators acting in this space.
Let us remind you that amount two linear operators A And IN called operator A + B, such that ( A + IN) X = Oh +In, X∈ L.
Product of linear operators A And IN is called a linear-linear operator AB, such that ( AB)X = A(In) for anyone X ∈ L.
Using the properties of the indicated operations on linear operators, we can show that the set of all linear operators acting in space L, together with the operations of addition and multiplication of operators, forms a ring. The zero of this ring is null operator, and by unit - identity operator.
This ring is called ring of linear operators in linear space L. #
The ring axioms are also called basic identities of the ring . A ring identity is an equality whose validity is preserved when any elements of the ring are substituted for the variables appearing in it. Basic identities are postulated, and from them other identities can then be deduced as consequences. Let's look at some of them.
Recall that the additive group of a ring is commutative and the operation is defined in it subtraction.
Theorem 2.8. In any ring the following identities hold:
- 0 ⋅ a = a ⋅ 0 = 0 ;
- (-a) ⋅ b = -(a ⋅ b) = a ⋅ (-b);
- (a-b) ⋅ c = a ⋅ c - b ⋅ c, c ⋅ (a-b) = c ⋅ a - c ⋅ b.
◀Let's prove the identity 0 ⋅ a = 0 . Let us write for arbitrary a:
a+ 0 ⋅ a = 1 ⋅ a + 0 ⋅ a = ( 1 +0 ) ⋅ a = 1 ⋅ a = a
So, a + 0 ⋅ a = a. The last equality can be considered as an equation in the additive group of a ring with respect to an unknown element 0 ⋅ a. Since in the additive group any equation of the form a + x = b has a unique solution x = b - a, then 0 ⋅ a = a - a = 0 . Identity a⋅ 0 = 0 is proved in a similar way.
Let us now prove the identity - (a ⋅ b) = a ⋅ (-b). We have
a ⋅ (-b)+a ⋅ b = a ⋅ ((-b) + b) = a ⋅ 0 = 0 ,
whence a ⋅ (-b) = -(a ⋅ b). In the same way, one can prove that (-a) ⋅ b = -(a ⋅ b).
Let us prove the third pair of identities. Let's consider the first of them. Taking into account what was proved above, we have
a ⋅ (b - c) = a ⋅ (b+(-c)) = a ⋅ b + a ⋅ (-c) =a ⋅ b - a ⋅ c,
those. the identity is true. The second identity of this pair is proved in a similar way.
Corollary 2.1. In any ring the identity ( -1 ) ⋅ x = x ⋅ ( -1 ) = -x.
◀The indicated corollary follows from the second identity of Theorem 2.8 for a = 1 and b = x.
The first two identities proved in Theorem 2.8 express a property called nullifying property of zero in the ring. The third pair of identities of this theorem expresses the distributive property of the operation of multiplication of a ring with respect to the operation of subtraction. Thus, when performing calculations in any ring, you can open the brackets and change the signs in the same way as when adding, subtracting and multiplying real numbers.
Nonzero elements a and b of the ring R called dividers zero , if a ⋅ b = 0 or b ⋅ a = 0 . An example of a ring with a zero divisor gives any modulo residue ring k if k is a composite number. In this case, the product modulo k of any type that yields a multiple of k during ordinary multiplication will be equal to zero. For example, in a residue ring modulo 6, elements 2 and 3 are zero divisors, since 2 ⨀ 6 3 = 0. Another example is given by a ring of square matrices of a fixed order (at least two). For example, for second-order matrices we have
When a and b are non-zero, the given matrices are zero divisors.
By multiplication, a ring is only a monoid. Let us pose the question: in what cases will a multiplication ring be a group? First of all, note that the set of all elements of the ring in which 0 ≠ 1 , cannot form multiplication groups, since zero cannot have an inverse. Indeed, if we assume that such an element 0" exists, then, on the one hand, 0 ⋅ 0" = 0" ⋅ 0 = 1 , and on the other - 0 ⋅ 0" = 0" ⋅ 0 = 0 , from which 0 = 1. This contradicts the condition 0 ≠ 1 . Thus, the question posed above can be refined as follows: in what cases does the set of all non-zero elements of a ring form a group under multiplication?
If a ring has zero divisors, then the subset of all non-zero elements of the ring does not form a multiplication group, if only because this subset is not closed under the multiplication operation, i.e. There are non-zero elements whose product is equal to zero.
A ring in which the set of all non-zero elements by multiplication forms a group is called body , commutative body - field , and the group of non-zero elements of the body (field) by multiplication - multiplicative group this body (fields). According to the definition, a field is a special case of a ring in which operations have additional properties. Let's write down all the properties that are required for field operations. They are also called field axioms .
The field is an algebra F = (F, +, ⋅, 0, 1), the signature of which consists of two binary and two nullary operations, and the identities are valid:
- a+(b+c) = (a+b)+c;
- a+b = b+a;
- a+0 = a;
- for every a ∈ F there is an element -a such that a+ (-a) = 0;
- a ⋅ (b ⋅ c) = (a ⋅ b) ⋅ c;
- a ⋅ b = b ⋅ a
- a ⋅ 1 = 1 ⋅ a = a
- for every a ∈ F different from 0, there is an element a -1 such that a ⋅ a -1 = 1;
- a ⋅ (b+c) = a ⋅ b + a ⋅ c.
Example 2.13. A. Algebra (ℚ, +, ⋅, 0, 1) is a field called field of rational numbers .
b. The algebras (ℝ, +, ⋅, 0, 1) and (ℂ, +, ⋅, 0, 1) are fields called fields of real and complex numbers respectively.
V. An example of a body that is not a field is algebra quaternions . #
So, we see that the field axioms correspond to the known laws of addition and multiplication of numbers. When doing numerical calculations, we “work in the fields,” namely, we deal primarily with the fields of rational and real numbers, sometimes we “move” to the field of complex numbers.
Definition 4.1.1. Ring (K, +, ) is an algebraic system with a non-empty set K and two binary algebraic operations on it, which we will call addition And multiplication. The ring is an Abelian additive group, and multiplication and addition are related by the laws of distributivity: ( a + b) c = a c + b c And With (a + b) = c a + c b for arbitrary a, b, c K.
Example 4.1.1. Let's give examples of rings.
1. (Z, +, ), (Q, +, ), (R, +, ), (C, +, ) – respectively, rings of integer, rational, real and complex numbers with the usual operations of addition and multiplication. These rings are called numerical.
2. (Z/nZ, +, ) – ring of residue classes modulo n N with addition and multiplication operations.
3. Many M n (K) all square matrices of fixed order n N with coefficients from the ring ( K, +, ) with the operations of matrix addition and multiplication. In particular, K may be equal Z, Q, R, C or Z/nZ at n N.
4. The set of all real functions defined on a fixed interval ( a; b) real number line, with the usual operations of addition and multiplication of functions.
5. Set of polynomials (polynomials) K[x] with coefficients from the ring ( K, +, ) from one variable x with natural operations of addition and multiplication of polynomials. In particular, polynomial rings Z[x], Q[x], R[x], C[x], Z/nZ[x] at n N.
6. Ring of vectors ( V 3 (R), +, ) with the operations of addition and vector multiplication.
7. Ring ((0), +, ) with addition and multiplication operations: 0 + 0 = 0, 0 0 = = 0.
Definition 4.1.2. Distinguish finite and infinite rings (according to the number of elements of the set K), but the main classification is based on the properties of multiplication. Distinguish associative rings when the multiplication operation is associative (points 1–5, 7 of example 4.1.1) and non-associative rings (point 6 of example 4.1.1: here ,). Association rings are divided into rings with one(there is a neutral element regarding multiplication) and without unit, commutative(the multiplication operation is commutative) and non-commutative.
Theorem4.1.1. Let ( K, +, ) is an associative ring with one. Then many K* invertible with respect to multiplication of ring elements K– multiplicative group.
Let's check the fulfillment of the definition of group 3.2.1. Let a, b K*. Let's show that a b K * .
(a b) –1 = b –1 A –1 K. Really,
(a b) (b –1 A –1) = a (b b –1) A –1 = a 1 A –1 = 1,
(b –1 A –1) (a b) = b –1 (A –1 a) b = b –1 1 b = 1,
Where A –1 , b –1 K– inverse elements to a And b respectively.
1) Multiplication in K* associatively, since K– associative ring.
2) 1 –1 = 1: 1 1 = 1 1 K* , 1 – neutral element with respect to multiplication in K * .
3) For a K * ,
A –1 K* , because ( A –1) a = a (A –1) = 1
(A –1) –1
=
a.
Definition 4.1.3. Many K* invertible with respect to multiplication of elements of the ring ( K, +, ) are called multiplicative ring group.
Example 4.1.2. Let us give examples of multiplicative groups of various rings.
1. Z * = {1, –1}.
2. M n (Q) * = G.L. n (Q), M n (R) * = G.L. n (R), M n (C) * = G.L. n (C).
3.
Z/nZ* – set of invertible classes of residues, Z/nZ * = {
| (k, n) = 1,
0 k < n), at n > 1
| Z/nZ * | =
(n), Where
– Euler function.
4. (0) * = (0), since in this case 1 = 0.
Definition 4.1.4. If in an associative ring ( K, +, ) with unit group K * = K\(0), where 0 is a neutral element with respect to addition, then such a ring is called body or algebra withdivision. The commutative body is called field.
From this definition it is obvious that in the body K* and 1 K* means 1 0, therefore the minimal body, which is a field, consists of two elements: 0 and 1.
Example 4.1.3.
1. (Q, +, ), (R, +, ), (C, +, ) – respectively numeric fields rational, real and complex numbers.
2. (Z/pZ, +, ) – a finite field from p elements if p– prime number. For example, ( Z/2Z, +, ) – the minimum field of two elements.
3.
A non-commutative body is quaternion body– set quaternions, that is, expressions of the form h=
a + bi + cj + dk, Where a,
b,
c,
d R,
i 2 =
= j 2 = k 2 = – 1,
i j= k= – j i,
j k= i= – k j,
i k= – j= – k i, with the operations of addition and multiplication. Quaternions are added and multiplied term by term, taking into account the above formulas. For everyone h 0 inverse quaternion has the form: 
.
There are rings with zero divisors and rings without zero divisors.
Definition 4.1.5. If the ring contains non-zero elements a And b such that a b= 0, then they are called zero divisors, and the ring itself – ring with zero dividers. Otherwise the ring is called ring without zero divisors.
Example 4.1.4.
1. Rings ( Z, +, ), (Q, +, ), (R, +, ), (C, +, ) – rings without zero divisors.
2.
In the ring ( V 3 (R), +, ) every non-zero element is a zero divisor, since 
for everyone 
V 3 (R).
3.
In the matrix ring M 3 (Z) examples of zero divisors are matrices
And
, because A B = O(zero matrix).
4.
In the ring ( Z/nZ, +, ) with composite n = k m, where 1< k,
m < n, residue classes 
And 
are zero divisors, since.
Below we present the main properties of rings and fields.
Brief description
Definition. A ring is an algebra K = ‹K, +, -, ·, 1› of type (2, 1, 2, 0), the main operations of which satisfy the following conditions:
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Ring. Definition. Examples. The simplest properties of rings. Homomorphism and isomorphism of rings.
Definition. A ring is an algebra K = ‹K, +, -, ·, 1› of type (2, 1, 2, 0), the main operations of which satisfy the following conditions:
- the algebra ‹K, +, -› is an Abelian group;
- the algebra ‹K, ·, 1› is a monoid;
- multiplication is distributive relative to addition, that is, for any elements a, b, c from K
(a + b) c = a c + b c, c (a + b) = c a + c b.
The main set K of the ring K is also denoted by |K|. The elements of the set K are called elements of the ring K.
Def. The group ‹K, +, -› is called the additive group of the ring K. The zero of this group, that is, the neutral element with respect to addition, is called the zero of the ring and is denoted 0 or 0 K.
Def. The monoid ‹K, ·, 1› is called the multiplicative monoid of the ring K. The element 1, also denoted by 1 K, which is neutral with respect to multiplication, is called the unit of the ring K.
A ring K is called commutative if a · b = b · a for any elements a, b of the ring. A ring K is called zero if |K| = (0 K).
Def. A ring K is called a domain of integrity if it is commutative, 0 K ≠ 1 K and for any a, b О K from a b = 0 it follows a = 0 or b = 0.
Def. Elements a and b of the ring K are called zero divisors if a ≠ 0, b ≠ 0 or ba = 0. (Any integrity region has no zero divisors.)
Example. Let K be the set of all real functions defined on the set R of real numbers. Sum f + g, product f g, function
f(-1) and the unit function 1 are defined: (f + g) (x) = f (x) + g(x);
(f g)(x) = f(x) g(x); (–f) (x) =–f (x); 1(x) = 1. Direct verification shows that the algebra ‹K, +, -, ·, 1› is a commutative ring.
The simplest properties. Let K be a ring. Since the algebra ‹K, +, -› is an Abelian group, then for any elements a, b, from K the equation b + x = a has a unique solution a + (-b), which is also denoted by a – b.
- if a + b = a, then b = 0;
- if a + b = 0, then b = -a;
- – (-a) = a;
- 0 · a = a · 0 = a;
- (-a)b = a(-b) = -(ab);
- (-a)(-b) = a · b;
- (a – b)c = ac – bc and c(a – b) = ca – cb.
Let K = ‹K, +, -, ◦, 1› and K` = ‹K`, +, -, ·, 1`› - rings. It is said that the mapping h of the set K to K` preserves the main operations of the ring K if the following conditions are met:
- h(a+b)=h(a)+h(b) for any a, b from the ring K;
- h(-a)=-h(a) for any a from K;
- h(a b) = h(a)◦h(b) for any a, b from K;
- h(1) = 1`.
Def. A homomorphism of the ring K into (on) the ring K` is a mapping of the set K into (on) K` that preserves all the main operations of the ring K. A homomorphism of the ring K onto K` is called an epimorphism.
Def. A homomorphism h of the ring K onto the ring K` is called an isomorphism if h is an injective mapping of the set K onto K`. Rings K and K` are said to be isomorphic if there is an isomorphism between the ring K and the ring K`.
Containing a unit is called ring with one . Unit is usually designated by the number “1” (which reflects the properties of the number of the same name) or sometimes (for example, in matrix algebra), by the Latin letter I or E.
Different definitions of algebraic objects may either require the presence of a unit or leave it as an optional element. A one-sided neutral element is not called a unit. The unit is unique in the general property of a two-sided neutral element.
Sometimes the units of a ring are called its invertible elements, which can cause confusion.
One, zero and category theory
The unit is the only element of the ring that is both idempotent and invertible.
Reversibility
Reversible Any element u of a ring with unity that is a two-sided divisor of unity is called, that is:
∃ v 1: v 1 u = 1 (\displaystyle \exists v_(1):v_(1)\,u=1) ∃ v 2: u v 2 = 1 (\displaystyle \exists v_(2):u\,v_(2)=1) (a 1 + μ 1 1) (a 2 + μ 2 1) = a 1 a 2 + μ 1 a 2 + μ 2 a 1 + μ 1 μ 2 1 (\displaystyle (a_(1)+\mu _( 1)(\mathbf (1) ))(a_(2)+\mu _(2)(\mathbf (1) ))=a_(1)a_(2)+\mu _(1)a_(2) +\mu _(2)a_(1)+\mu _(1)\mu _(2)(\mathbf (1) ))while maintaining such properties as associativity and commutativity of multiplication. Element 1 will be the unit of extended algebra. If there was already a unit in the algebra, then after expansion it will turn into an irreversible idempotent.
This can also be done with a ring, for example, because every ring is an associative algebra over
Let (K,+, ·) be a ring. Since (K, +) is an Abelian group, taking into account the properties of groups we obtain
SV-VO 1. In every ring (K,+, ·) there is a unique zero element 0 and for every a ∈ K there is a unique element opposite to it -a.
NE-VO 2. ∀ a, b, c ∈ K (a + b = a + c ⇒ b = c).
SV-VO 3. For any a, b ∈ K in the ring K there is a unique difference a − b, and a − b = a + (−b). Thus, the subtraction operation is defined in the ring K, and it has properties 1′-8′.
SV-VO 4. The multiplication operation in K is distributive with respect to the subtraction operation, i.e. ∀ a, b, c ∈ K ((a − b)c = ac − bc ∧ c(a − b) = ca − cb).
Doc. Let a, b, c ∈ K. Taking into account the distributivity of the operation · in K with respect to the operation + and the definition of the difference of elements of the ring, we obtain (a − b)c + bc = ((a − b) + b)c = ac, whence by definition difference it follows that (a − b)c = ac − bc.
The right law of distributivity of the multiplication operation relative to the subtraction operation is proved in a similar way.
SV-V 5. ∀ a ∈ K a0 = 0a = 0.
Proof. Let a ∈ K and a b-arbitrary element from K. Then b − b = 0 and therefore, taking into account the previous property, we obtain a0 = a(b − b) = ab − ab = 0.
It is proved in a similar way that 0a = 0.
NE-VO 6. ∀ a, b ∈ K (−a)b = a(−b) = −(ab).
Proof. Let a, b ∈ K. Then (−a)b + ab = ((−a) + a)b =
0b = 0. Hence, (−a)b = −(ab).
The equality a(−b) = −(ab) is proved in a similar way.
NE-VO 7. ∀ a, b ∈ K (−a)(−b) = ab.
Proof. Indeed, applying the previous property twice, we obtain (−a)(−b) = −(a(−b)) = −(−(ab)) = ab.
COMMENT. Properties 6 and 7 are called the rules of signs in the ring.
From the distributivity of the multiplication operation in the ring K relative to the addition operation and properties 6 and 7, the following follows:
SV-VO 8. Let k, l be arbitrary integers. Then ∀ a, b ∈ K (ka)(lb) = (kl)ab.
Subring
A subring of a ring (K,+, ·) is a subset H of a set K that is closed under the operations + and · defined in K and is itself a ring under these operations.
Examples of subrings:
Thus, Z is a subring of the ring (Q,+, ·), Q is a subring of the ring (R,+, ·), Rn×n is a subring of the ring (Cn×n,+, ·), Z[x] is a subring of the ring ( R[x],+, ·), D is a subring of the ring (C,+, ·).
In any ring (K,+, ·), the set K itself, as well as the singleton subset (0) are subrings of the ring (K,+, ·). These are the so-called trivial subrings of the ring (K,+, ·).
The simplest properties of subrings.
Let H be a subring of the ring (K,+, ·), i.e. (H,+, ·) is itself a ring. This means that the (H, +)-group, i.e. H is a subgroup of the group (K, +). Therefore, the following statements are true.
SV-VO 1. The zero element of the subring H of the ring K coincides with the zero element of the ring K.
SV-VO 2. For any element a of the subring H of the ring K, its opposite element in H coincides with −a, i.e. with its opposite element in K.
SV-VO 3. For any elements a and b of the subring H, their difference in H coincides with the element a − b, i.e. with the difference of these elements in K.
Signs of a subring.
THEOREM 1 (first sign of a subring).
A non-empty subset H of a ring K with the operations + and · is a subring of the ring K if and only if it satisfies the following conditions:
∀ a, b ∈ H a + b ∈ H, (1)
∀ a ∈ H − a ∈ H, (2)
∀ a, b ∈ H ab ∈ H. (3)
Necessity. Let H be a subring of the ring (K,+, ·). Then H is a subgroup of the group (K, +). Therefore, by the first criterion of a subgroup (in the additive formulation), H satisfies conditions (1) and (2). Moreover, H is closed under the multiplication operation defined in K, i.e. H
also satisfies condition (3).
Adequacy. Let H ⊂ K, H 6= ∅ and H satisfies conditions (1) − (3). From conditions (1) and (2) according to the first criterion of a subgroup it follows that H is a subgroup of the group (K, +), i.e. (H, +)-group. Moreover, since (K, +) is an Abelian group, (H, +) is also Abelian. In addition, from condition (3) it follows that multiplication is a binary operation on the set H. The associativity of the operation · in H and its distributivity with respect to the operation + follow from the fact that the operations + and · in K have such properties.
THEOREM 2 (second sign of a subring).
A non-empty subset H of a ring K with the operations + and · is
subring of the ring K t. and t. t, when it satisfies the following conditions:
∀ a, b ∈ H a − b ∈ H, (4)
∀ a, b ∈ H ab ∈ H. (5)
The proof of this theorem is similar to the proof of Theorem 1.
In this case, Theorem 2′ (the second criterion of a subgroup in the additive formulation) and a remark to it are used.
7.Field (definition, types, properties, characteristics).
A field is a commutative ring with identity e is not equal to 0 , in which every element different from zero has an inverse.
Classic examples of number fields are the fields (Q,+, ·), (R,+, ·), (C,+, ·).
PROPERTY 1 . In every field F the law of contraction is valid
by a common factor different from zero, i.e.
∀ a, b, c ∈ F (ab = ac ∧ a is not equal to 0 ⇒ b = c).
PROPERTY 2 . In every field F no zero divisors.
PROPERTY 3 . Ring(K,+, ·) is a field if and only
when there are many K\(0) is a commutative group with respect to the operation of multiplication.
PROPERTY 4 . Finite nonzero commutative ring(K,+, ·) without zero divisors is a field.
The quotient of the field elements.
Let (F,+, ·) be a field.
Partial elements a And b fields F , Where b is not equal to 0 ,
such an element is called c ∈ F , What a = bc .
PROPERTY 1 . For any elements a And b fields F , Where b is not equal to 0 , there is a unique quotient a/b , and a/b= ab−1.
PROPERTY 2 . ∀ a ∈ F \ (0)
a/a= e And∀ a ∈ F a/e= a.
PROPERTY 3 . ∀ a, c ∈ F ∀ b, d ∈ F \ (0)
a/b=c/d ⇔ ad = bc.
PROPERTY 4 . ∀ a, c ∈ F ∀ b, d ∈ F \ (0)
PROPERTY 5 . ∀ a ∈ F ∀ b, c, d ∈ F \ (0)
(a/b)/(c/d)=ad/bc
PROPERTY 6 . ∀ a ∈ F ∀ b, c ∈ F \ (0)
PROPERTY 7 . ∀ a ∈ F ∀ b, c ∈ F \ (0)
PROPERTY 8 . ∀ a, b ∈ F ∀ c ∈ F \ (0)
Field F , whose unit has finite order p in the group(F, +) p .
Field F unit, which has infinite order in the group(F, +) , is called the characteristic field 0.
8. Subfield (definition, types, properties, characteristics)
Field subfield(F,+, ·) called a subset S sets F , which is closed under the operations+ And· , defined in F , and itself is a field relative to these operations.
Let us give some examples of subfields Q-subfield of the field (R,+, ·);
R-subfield of the field (C,+, ·);
The following statements are true.
PROPERTY 1 . Subfield element zero S fields F coincides with
zero element of the field F .
PROPERTY 2 . For every element a subfields S fields F its opposite element in S coincides with−a , i.e. with its opposite element in F .
PROPERTY 3 . For any elements a And b subfields S fields F their
difference in S coincides with a−b those. with the difference of these elements in F .
PROPERTY 4 . Subfield unit S fields F coincides with one
e fields F .
PROPERTY 5 . For every element a subfields S fields F , from-
personal from zero, its inverse element in S coincides with a−1 , i.e. with the element inverse to a V F .
Signs of the subfield.
THEOREM 1 (the first sign of a subfield).
Subset H fields F with operations+, · , containing non-zero
(F,+, ·)
∀ a, b ∈ H a + b ∈ H, (1)
∀ a ∈ H − a ∈ H, (2)
∀ a, b ∈ H ab ∈ H, (3)
∀ a ∈ H \ (0) a−1 ∈ H. (4)
THEOREM2 (second sign of the subfield).
Subset H fields F with operations+, · , containing non-zero
element is a subfield of the field(F,+, ·) if and only if it satisfies the following conditions:
∀ a, b ∈ H a − b ∈ H, (5)
∀ a ∈ H ∀ b ∈ H\(0) a/b ∈ H. (6)
10. Divisibility relation in the ring Z
Statement: for any elements a,b,c of a commutative ring on the set R, the following implications hold:
1) a|b, b|c => a|c
2) a|b, a|c => a| (b c)
3) a|b => a|bc
for any a, b Z the following is true:
2) a|b, b≠0 => |a|≤|b|
3)a|b and b|a ó |a|=|b|
Dividing the integer a by the integer b with the remainder means finding integers q and r such that you can represent a=b*q + r, 0≤r≥|b|, where q is the incomplete quotient, r is the remainder
Theorem: If a and b Z, b≠0, then a can be divided by b with a remainder, and the incomplete quotient and remainder are uniquely determined.
Corollary, if a and b Z , b≠0, then b|a ó
11. GCD and NOC
The greatest common divisor (GCD) of the numbers Z is some number d that satisfies the following conditions
1) d is a common divisor i.e. d| ,d| …d|
2) d is divisible by any common divisor of numbers, i.e. d| ,d| …d| =>d| ,d| …d|
