II Fundamentals of General Algebra

be arbitrary numbers ; then there exists for every pair of them a uniquely determined number s, the sum

Although these formulae are fundamental for calculation with numbers, they do not characterize completely the notion of number. Of course, in the different branches of mathematics, the word number is used in different senses. For example, you speak of natural, rational, complex, hyper-complex numbers, etc. The rational numbers as well as the real numbers and the complex numbers form three systems each of which satisfies the above conditions, whereas the system formed by the natural numbers does not satisfy the laws of inverse existence; the reader may verify this statement!

On the other hand, in the system formed by all the analytic functions of a complex variable, all the laws mentioned before apply, although the system is not composed of numbers. In this chapter, systems of any kind will be considered, where either all these laws or some of them hold. In general, nothing will be assumed about the nature of the mathematical elements which form these systems. It is essential that the elements are interconnected with the aid of certain operations which obey particular laws. These operations will sometimes be called rational operations.

2.1.2 Modules: A system of elements, for which an operation satisfying the conditions (4a), (5a), (6a) of 2-11 is defined, is said to form an Abelian group or a module*. The system formed by the rational numbers is an instance of a module; so is the system of the real (complex) numbers. Vector spaces (cf. 1.3) are modules of a different type; their elements are n-vectors, and addition, but not multiplication of n-vectors has been defined. Consider especially n = 3. The 3-vectors can be represented by vectors in space; one gets in this way modules which are formed by geometrical entities (vectors). Moreover, there corresponds to every vector a parallel displacement of the space transforming the starting point of the vector into its end-point, and conversely there corresponds to every parallel displacement of the space one and only one vector, to the sum of two vectors a + b the parallel displacement which is generated by performing successively the two parallel displacements corresponding to a and to b. Thus, the parallel displacements of space form a module; this module is therefore composed of elements which are geometrical transformations.

* There is no essential difference between the meaning of the two terms. The term addition and the sign + are only notations, and there is no harm done in replacing them by other words. In these cases, it is unusual to speak of modules and the word module is replaced by Abelian group.

The integral numbers form a module, in which, besides addition and subtraction, also multiplication can be performed, while division is only possible in special cases.

Rotations about any particular axis form a module which is connected to the module of the real numbers as follows: Let g be a particular positive integral number and let there correspond to every real number a the rotation by the angle 2pa/g; then there corresponds a rotation to every real number and to two real numbers a and b the same rotation, if and only if a - b is an integral multiple of g. Two rotation corresponding to the numbers k and l generate, when taken one after the other, a rotation which corresponds to k + l. Rotations for which a is an integral number, form a finite system which is a module. Each of these rotations corresponds to one of the integral numbers 0, 1, 2, ··· , g - 1, which occur as residues when an integral number is divided by g. Two integral numbers corresponding to the same rotation (and therefore to the same residue) are said to be congruent:

they form a class of residues modulo g. It will be proved later on that these classes form a module ; this module is of pecial interest, especially in the case when g is a prime number.

2.1.3 Partition into classes: In the example considered above, a partition of the set of all integers into classes has been generated by a congruence of its elements. This concept will now be generalized.

An arbitrary set A of elementa a, b, c, ··· may be decomposed into classes so that every element belongs to one and only one class. Two elements are said to be equivalent, written a ~ b, if they belong to the same class. Then equivalence has the properties:

a ~ a	Law of reflexivity
If a ~ b, then b~ a	Law of symmetry	(1)
If a ~ b, If b ~ c, then If a ~ c	Law of transitivity

Hence there corresponds to every partition of a set A into classes an equivalence of its elements such that the three laws (1) above are satisfied for this equivalence. However, the ordinary way of mathematical investigation is the converse one: An equivalence between the elements of A is given and from this equivalence a partition into classes is derived. It has been shown just before that an equivalence which generates a partition into classes must satisfy the conditions (1); the following lemma establishes that these conditions are not only necessary, but also are sufficient for a partition into classes:

Lemma: Given an equivalence between the elements of A satisfying the conditions (1), and letting (a), (b), (c), ··· be the classes of the elements equivalent to a, b, c, ···, respectively, then each element of A. belong to class and two classes have either all elements in common - i.e., they are' identical — or they have no common element.

Proof: As a ~ a, the element a itself belongs to the class (a), formed by the elements equivalent to (a). If b is a common element of (a) and (c), it follows from the law of. symmetry that a and c belong to (b). By the law of transitivity, each element of (a) and of (c) belongs to (b), and each element of (b) belongs to (a) and to (c). Hence (a), (b) and (c) are identical.

Each element of a class will be called its representative with the notation:

The method of forming classes often creates new mathematical entities. Operations on classes are derived from operations on original elements as follows:

Let an operation, say addition, exist for the elements of any system A and let there be given a partition of A into classes. The sum of two classes is mostly defined by

But this definition is admissible if and only if the class (c + d) is the same whatever elements c and d are chosen as representatives of their classes. The operation for multiplication is similar.

Now the congruence (mod g), defined in 2.1.2, is an equivalence satisfying obviously the conditions (1). This congruence generates a partition into g classes of residues

The above lemma will now be applied to these classes. As every integer is congruent to its residue after division by g, (a) = (a') if and only if a º a' (mod g) ; moreover, if (a) = (a'), (b) = (b'), a' = a + rg, b' = b + sg, whence (a + b) and (a' + b') = (a + b + [r + s] g) denote the same class. It is therefore admissible to define addition and multiplication of classes by

(a) + (b) = (b) + (a),	[(a) + (b)] + (c) = (a) + [(b) +(c)],	(a) + (b - a) = (b),
(a)(b) = (b)(a),	[(a)(b)] (c) = (a)[(b)(c)],
[(a) + (b)](c) = (a)(c) + (b)(c),	(a)[(b) + (c)] = (a) (b) + (a)(c).

Hence the classes of residues (mod g) form a module in which a second operation, multiplication satisfying the conditions' (4m), (5m) and (7) of 2.1.1 is defined.

2.1.4 Singular elements: Although a module is not necessarily a set of numbers, there exists in every module an element which has about the same properties as the number zero and which is therefore used to be denoted by the character 0.

Theorem: There exists in a module M one -and only one singular element 0, such that for every element a of M, a+ x equals (does not equal) a, if x is the singular (a non-singular) element.

Proof: Let a be any particular element of M. As the law of inverse existence (6a) of 2.1.1 holds for addition, as defined in M, there must exist an element, say 0 of M, satisfying the condition

Thus the sum of a and 0 is a itself. It will be proved now that this element 0 has the same property with respect to every element of M and that it is unique. Let b be an arbitrary element of M, then there exists an element c of M, satisfying

In order to prove its uniqueness, it may be assumed that there exists another element, say 0 in M, so that a + 0 = a, when 0 has the same properties as 0 and the elements of M will not change, when 0 is added. Hence 0 + 0 = 0, but on the other hand 0 + 0 = 0, whence 0 = 0.

2.1.5 Operations in a module: The addition of a' is the operation inverse to the addition of a. The addition of a' will therefore be called the subtraction of a and the following notation used:

Now (a₁ + a₂) + a₃ = a₁ + (a₂ + a₃), whence one can omit the brackets in this sum. By mathematical induction, it can be proved in the same manner, as it is done in elementary arithmetic, that the brackets in the sums of n elements can be omitted. In place of a₁ + a₂ + ··· + a_n), one writes sometimes

this notation is often further abbreviated by replacing

by Sa_i whenever no ambiguity can arise. If a₁ = a₂ = ··· = a_n= a, the sum will be denoted by

The product of a non-positive integer with an arbitrary element a of M is defined by

A system S', formed by elements of any set S, is said to be a sub-set of S. In particular, a subset M' of a module (an Abelian group) M is said to be a sub-module (a sub-group) of M, if it forms a module with respect to the addition defined in M.

Theorem: A subset M ' is a sub-module of M if and only if the differences of the elements of M' belong to M'.

Proof: This condition is obviously necessary. Let the condition hold and let a and b be elements of M', then b - b = 0, 0 - b = - b and a - (-b) = a + b belong to M'. Hence the addition defined in M can be carried out in M'; for this addition, the associative and the commutative laws hold and the equation x+a=b can be solved by the element b - a of M'. Thus, M' is a sub-module of M.

2.1.6. Rings: If in a module T, multiplication satisfying the associative and distributive laws is defined in such a manner that the product of every pair of elements belongs to T, this module is called a ring, and if in a ring multiplication satisfies the commutative law (4m) of 2.1.1, the ring is said to be a commutative ring.

The integral numbers form a commutative ring R; other instances of commutative rings are the sets R_g of the integral multiples of an integral number g and the classes of residues (mod g). Non-commutative rings are, for example, formed by matrices (cf. Chapter VI)

A subset of a ring T, which itself is a ring with the addition and the multiplication, as defined in T as its operations, is said to be a sub-ring of T.

(1) A subset T' of a ring T is a sub-ring of T if and only if the differences and the products of the elements of T' belong to T'.
(2) Replace the law of reflexivity by the condition: For every element of A, there exists at least one element which is equivalent to it, and show that this condition in connection with the laws of symmetry and transitivity implies the law of reflexivity.

Hence it is impossible to satisfy Condition (6m) of 2.1.1 in any ring without restriction. If you want to introduce the condition that to every pair of elements a, b there should exist a y such that

You obtain this restriction from Equation (6o) of 2.1.1 by replacing the number 0 by the singular element 0 of the module T; when mentioned in the sequel, the formula (6o) of 2.1.1 should always be understood in this sense.2.2 Fields: If in a commutative ring, which contains more than one element, the condition (6m) with the restriction (6o) of 2.1.1 holds, it is said to be a field. In other words: A field is a set of more than one element in which an addition and a multiplication are defined, and the conditions (4a), (4m), (5a), (5m), (6a), (6m) with (6o) and (7) of 2.1.1 hold for these operations.

Theorem: If a, b are elements of a field F and a b = 0, then at least one of the factors a, b must be equal to 0.

Proof: Let a ¹ 0, b ¹ 0. By (6m) and (6o) of 2.1.1, there is in F an element c such that cb ¹ 0 (for example, cb = a) and an element y such that ya = c. Then 0 = y0 = yab = cb ¹ 0, whence follows the theorem.

By this theorem, the elements ¹ 0 of a field F form a system S, where multiplication is commutative and associative and in which for every pair a,b of elements of S the equation ax = b has a solution in S. Hence multiplication satisfies in S the same conditions as addition must satisfy in a module; only the sign of + has been replaced by the notation of multiplication. Hence you can translate the results of 2.1.6 from the language of addition into the language of multiplication.

Theorem 1: There is one and only one element 1 in a field satisfying (1) for every element a of the field.

By translating the theorem of 2.1.5 into the language of multiplication , you obtain

Theorem 2: If a and b are elements of a field F and a ¹ 0, then the equation ax = b has one and only one solution in F, namely, x = a^-1b.

Corresponding to the sum of n elements, one can form the product

of n elements, and if all of the elements are equal, the product is the power

The necessary and sufficient conditions, which a field must satisfy, can also be given in the following manner:

A system F of more than one element, for which addition and multiplication are uniquely defined, is a field if and only if:

1. The elements form a module; the singular element may be denoted by 0.
2. Elements different from 0 form a system S of at least one element which is an Abelian group with respect to multiplication.
3. a 0 = 0 a = 0.
4. a(b + c) = ab + ac for arbitrary elements a, b, c of F.

The singular element 0 of the module will be called the null element or zero element.

of & set A and an addition and a multiplication exist in A for which the formulae

are satisfied for all elements a, b of T, then the representation is said to be a homorphism and A is said to be homomorphic to T. The element a(a) of A is said to be the image of the original element a of T. The representation of a by a(a) is denoted sometimes by

one says that a particular homomorphism maps A onto T. There exit homomorphisms where different elements are mapped onto the same image.

Proof: Every element of A is of the form a(a), where a is a suitable element of T, not necessarily defined uniquely by its image a(a).

[a(a) + a(b)] +a(c) = a(a + b + c) = a(a) + [a(b) + a(c)],
a(a) + a(b) = a(a + b) = a(b) + a(a) ,		(2)
a(a) + a(b - a) = a(b).

After replacing the notion of addition in (2) by that of multiplication, you realize that multiplication in A is always associative, and that it is commutative if T is a commutative ring. In order to verify the distributive laws, consider

The second distributive law follows from this formula and the commutative law and hence follows theorem.

Exercise: Prove: a(aⁿ) = a(a)ⁿ for any positive integer n and a(ma) = m a(a), for any integral number m.

a(b) + a(-b) = a(0),		whence
a(-b) = -a(b),		and therefore
a(a - b) = a(a) + (-b) = a(a) - a(b),		whence follows

Theorem 2: The necessary and sufficient condition for a(a) = a(b) is a(a-b)=a(0).

The number of the original elements which are mapped onto a(0) is important as is seen from the

Corollary: Either every element of A is an image of more than one element of a ring T, or every element of A is an image of one and only one element of T.

In the second case, A is said to be isomorphic to T, and the homomorphism is called isomorphism. An isomorphism is therefore a (1,1)-correspondence of two rings by which sums, differences and products of corresponding pairs of elements correspond. Hence, in certain cases, isomorphic rings are considered to be equal or to differ only by the notation of the elements, while, in general, the notion of isormorphism must be distinguished from that of identity. For example, a ring T may have two different sub-rings which are isomorphic. Of special interest is the case when two sets are isomorphic and form the same set: This isomorphism. is called an automorphism - a permutation of elements for which addition and multiplication are invariant. The isomorphism, which maps A onto B and conversely is sometimes denoted by A ¬ ® B.

1. The ring of the classes of residues (mod g) is homomorphic to the ring of the integers.
2. In the ring of the numbers a + bÖ2 (a, b integers), the transformation
a + bÖ2 ® a - bÖ2 is an automorphism.

2.2.3. Ring of classes of residues generated by a homomorphism: If a is a homomorphism of a ring T, there is a partition of the elements of T into classes, two elements a and b being equivalent if a(a) = a(b), i.e., if a(a-b)=a(0). These classes are classes of residues as considered in 2.1.3, but they are of a particular kind as the elements c for which a(c) = a(0) form a sub-ring T' of T with the property that each product of an element of T and an element of T' belongs to T". The converse of this statement is contained in

Theorem 1: Let a ring T contain a sub-ring T' and let every product in T, one factor of which is an element of T', be contained in T', then the classes of residues of T' in T form a ring which is homomorphic to T.

Proof: Let r, r', ··· be elements of T' and consider two elements of T to be equivalent if and only if their difference belongs to T'. This definition satisfies the conditions of 2.1.3 and furnishes therefore a partition of T into classes. Moreover,

Hence it is admissible to define the addition and multiplication of the classes (a), (b), ··· by the corresponding operations exercised on their representatives

Every rational formula, which is true for elements a, b, ···, remains so if these elements are replaced in the formula by the classes which they represent. Hence the classes form a module in which there hold the associative law of multiplication and the distributive law, i.e., they form a ring which is homomorphic to T.

For example, consider the rings which are homomorphic to the ring R of the integral numbers. In order to investigate this important class of rings, one has to find out all the sub-rings of R with the desired property. Every sub-ring is also a sub-module; in the case of R, it is easy to prove that also the converse is true.

Lemma: Every sub-module of the ring R of integral numbers is a ring R_g consisting of the multiples of a non-negative integral number g.

Proof: If ± g is contained in a sub-module M of R, then M contains R_g. Now M either consists of 0 only and is therefore equal to R, or it contains a minimum positive integer, say g. Let m be any number out of M and m=kg+g', where g' is the residue 0 £ g' < g. Then g' is an element of M, and since g is the smallest positive element of M, g' = 0. Hence m is contained in R_g and therefore M = R_g.

Obviously, R_g is a commutative ring and has especially the property, assumed in the preceding theorem, that the product of any element of B and an element of R_g belongs to R_g.

In order to obtain the rings which are homomorphic to R, one has to consider the classes of residues of R_g in R (for g = 0, 1, ···).

Let g = 0, then R_g consists of the number 0 only. Every class of residues consists of one element only. The ring formed by these classes is therefore isomorphic to R.

Let g = 1, R₁ is identical to R. There is one class of residues only, all the numbers are equivalent to 0. The ring formed by the classes of residues consists of the null-element only.

Let g > 1. The classes of residues form a commutative ring G consisting of the elements

which has already been considered in 2.1.3. G is homomorphic, but obviously not isomorphic to R. One has to make a distinction between whether g is prime or not. This distinction is afforded by the following two theorem:

Theorem 2: If g > 1 is not a prime number, the classes of residues of g form a commutative ring G which itself is neither a field nor a sub-ring of any field.

Proof: It follows directly from the Theorem 1 that G is a commutative ring. As g > 1 is not prime, g = r s, where both the factors are positive and less than g. The classes of residues (r) and (s) both differ from (0), but their product (r)(s)=(0). Since in a field the product of two elements, which are different from the null-element, itself differs from the null-element, G cannot be a field nor a sub-ring of a field.

Proof: In order to prove the theorem, it must only be shown that for every particular pair of classes (a) ¹ (0) and (b), the equation

has a solution. Let (x) run over the p classes (1) and consider the products (a)(x). If two of them are equal, say (a) (x₁ - x₂) = (0), then (a) (x₁ - x₂) = (0), and therefore a [x₁ - x₂] is divisible by p. However, since for (a) ¹ (0), a is not divisible by p, the factor x₁ - x₂ must be divisible by p. Hence (x₁ = (x₂). The p products (a) (x) form therefore p different classes. One of them must be the class (b); Equation (2) has therefore a solution. Hence follows the theorem.

It has been shown by this theorem that a field may be homomorphic to a ring which itself is not a field. One may wonder whether a ring can be homomorphic to a field; it is of special interest to know whether a field can be homomorphic to a field without being isomorphic to it.

Theorem 4: When a ring A with more than one element is homomorphic to a field F, then A is homomorphic to F and A is therefore a field.

Proof: :Let a, b, c, d, ··· denote elements of F. If a and b are represented by the same element of A, then a - b is represented by the null-element. Let c be represented by an element a of A, which differs from the null element, and put c/(a—b) = a. The product d(a— b) must be represented by the null-element, since a - b is represented by it; this contradicts the assumption that d(a - b)=c is represented by a. Hence any two different elements of F must be represented by different elements of A, that is, A is isomorphic to F, whence it is a field.

A ring can therefore be homomorphic to a field only in the two trivial cases when the ring either consists solely of a null-element or is isomorphic to the field. As a field is supposed to contain more than one element, this yields the

Corollary: If a field is homomorphic to another field, it is isomorphic to it.

be modules, finite or infinite in number. If these modules have common elements, they form a module, which is called the meet of modules (1), since, if a and b are common elements, then a - b belongs to M₁ as well as to M₂, ··· and is therefore a common element. In the special cases when the modules are rings, the products ab belong also to the meet which is therefore a ring. If the modules are fields and the meet contains more than one element, the meet itself is a field. A sub-ring F' of a field F, which itself is a field, is said to be a sub-field of F ; this connection is denoted by F' Í F and, if F' ¹ F, by F'ÌF. If a subset X of F has the property that the sum, the difference, the product and the quotient of any two elements of X (the divisor being ¹ 0) belong to Z, then X is a sub-field of F. Indeed, the commutative, associative and distributive laws hold in X as they hold in F. The meet of all sub-fields of a field F contains at least the elements 0 and 1, whence it is a field, called the prime-field of F. The prime-field of F is therefore a sub-field of F as well as a sub-field of every sub-field of F; it has no other sub-field than itself and is the only sub-field with this property.

Theorem: The elements, obtained by repeated addition, sub-traction, multiplication and division of the unit-element of F and of the elements generated in this manner, form the prime-field of F.

Proof: Every sub-field of F must contain an element different from the null-element, say, the element a. Moreover, it contains a - a = 0, a/a = 1 and all those elements which are generated by repeated addition, subtraction, multiplication and division of 1 and of the elements generated successively by these operations. Hence these elements form a set P which is contained in the prime-field of F. On the other hand, the sums, differences, products and quotients—unless the divisor is the null-element - of two elements of P belong to P. Hence P is a sub-field of the prime-field and P is therefore itself the prime-field.

It follows from this theorem that the prime-field of the field of the complex numbers is the field of the rational numbers, and that the fields GF_p are their own prime-fields.

2.2.5 Prime-fields: In order to investigate the prime-field of F, consider the module generated by the unit-element 1 of F. This module is a sub-module of this prime-field and consists of the elements

use mathematical induction. The formula is obvious for q = 0 and every arbitrary integral value of p. It follows from

thus (2) is correct also for q + 1 and q - 1. Hence (1) holds for every pair of integral numbers p, q. The module formed by the elements (1) is therefore a ring R* and is homomorphic to the ring of the integral numbers. Hence there is no harm done by introducing the shorter notation

p +q denotes the element of F which corresponds to p + q, and it is simultaneously the sum of p and q; the corresponding result holds for p — g and p g. Note that the Italic characters

denote integral numbers which may not be elements of F, On the other hand, the elements (4) may not be different.

For example, let F be a field GF_p [cf. 2.2.3, Theorem 2], say GF₅,,then 2=7, etc. The notation 0 for the null-element agrees with the notation of (2.1), The ring of the elements (4) is homomorphic to the ring of the integral numbers and is therefore isomorphic to a ring of classes of residues. The null-element corresponds to a sub-ring G_g of the ring of the integral numbers [cf.2.2.3]. The number g is called the characteristic of the field F.

2.2.6 Fields of Characteristics p: Let g > 0. Then R* is isomorphic to the ring G, as considered in 2.2.3. It follows from 2.2.3 Theorem 2 that g is a prime number p and R* is therefore homomorphic to GF_p.

Hence R* is a field and since it is contained in the prime field which has no sub-field different from itself, R* is the prime-field of F. Hence

Theorem: If the characteristic of a field F is different from zero, it is a prime-number p. The prime-field consists of the elements

That every prime-number is indeed the characteristic of a suitably chosen field is obvious by the example of the fields GF_p. In a field of a characteristic 2, a=-a + 2a = - a holds: the positive sign and the negative sign are therefore not different. The calculation in fields of characteristic 2 is much simplified by this fact; on the other hand, there exists no arithmetic mean of two elements. For this reason, the case of fields of characteristic 2 has sometimes to be considered separately.

are divisible by p for 0 < m < p, since the prime factor p does not occur in the denominator. If one calculates in a field of characteristic p, those numbers have therefore to be replaced by zero. Hence in a field p, one has

holds for every odd prime number p; the same formula holds also for p = 2, since in fields of characteristic 2 subtraction does not differ from addition. Thus (2) is true for every field of characteristic p.

2.2.7 Fields of Characteristic 0: Let g = 0; then R* is isomorphic to R and therefore it cannot be a field; in order to discover its nature, apply the following theorem which holds for fields of any characteristic:

Theorem 1: Let A be a sub-ring of a field F and contain more than one element; the meet of those sub-fields of F which contain A is a field consisting of the solutions X of the equations

where a ¹ 0 and b are elements of A. In other words: The meet of the sub-fields is proposed to consist of the quotients of the elements of A.

Proof: The meet of those sub-fields is a field containing the set X of the solutions. One must therefore prove only that X itself is a field. Since A is supposed to contain an element a ¹ 0 and a1 = a, a 0 = 0, the elements 0 and 1 belong to X. Therefore X contains more than one element. We only need to show that the sum, difference, product and quotient of any two elements of X (the divisor being assumed ¹ 0) belong to X. Let a₁x₁ = b₁, a₂x₂ = b₂, then

a₁a₂(x₁ ± x₂) = a₂b₁ ± a₁b₂, (1)
a₁a₂(x₁x₂) = b₁ b₂, (2)

The meet P of all the sub-fields of F which contain R* is a sub-field of F and contains therefore the prime-field; on the other hand, the prime-field is one of the sub-fields of F containing R*, whence P is the prime-field of F. If F is supposed to be of characteristic zero, there corresponds to every pair of integral numbers r and s ¹ 0 a pair of elements r and s ¹ 0 in R^* and there exists therefore in P a quotient r/s. It follows from (1), (2) and (3) that addition, subtraction, multiplication and division of those quotients are done in the same manner as the corresponding operations for rational numbers. Hence P is homomorphic to the field of rational numbers. A field cannot be homomorphic to a field unless the two fields are isomorphic. P is therefore isomorphic to the field of rational numbers, whence follows

Theorem 2: The prime-field of a field of characteristic 0 is isomorphic to the field of rational numbers.

2.2.8 Quotient fields: The methods used in 2.2.7 will now be applied to characterize those rings which are sub-rings of a field. As in every field multiplication is commutative, a ring which is a sub-ring of a field must be a commutative ring; moreover, as the product of two elements which differ from 0 differs itself from 0, the same holds in every sub-ring of a field. It will be shown that these two necessary conditions are also sufficient.

Theorem: Let A be a commutative ring with the property that any pair of its elements which differ from zero form a product which differs from zero. Then A generates a field which is called the quotient-field Q(A) of A. The field Q(A) contains a sub-ring A' which is isomorphic to A and every element of Q(A) is a quotient of two elements of A'.

Proof: Consider the pairs of elements a, b of A for which b ¹ 0 . These pairs are distributed into classes with the aid of the following equivalence :

if there exist elements c and d of A such that c ¹ 0, d ¹ 0, ca = da' and cb=db'. Obviously, this equivalence is reflexive and symmetric. In order to prove the transitivity, assume that a', b' ~ a", b", whence c'a' = d'a", c'b'=d'b", where c' ¹ 0, d' ¹ 0. Then cc' a = dd' a", ce' b = dd' b". Since cc' ¹ 0, dd'¹0, a,b ~ a", b". Thus the equivalence generates a partition of the pairs into classes. The class represented by a, b will be denoted by a/b.In particular, a/b = ca/eb. Addition and multiplication of classes will now be defined by the following formulae (well known from the calculation with fractional numbers):

a₁/b₁ + a₂/b₂ = (a₁b₂+ a₂/b₁)/b₁b₂, (1)
a₁/b₁ a₂/b₂ = a₁a₂/b₁b₂. (2)

It must be proved that these formulae do not depend on the choice of the representatives.

a₁a₂/b₁b₂ = c₁c₂a₁a_2//c₁c₂b₁b₂= d₁d₂a'₁a'₂/d₁d₂b'₁b'₂ = a'₁a'₂/b'₁b'₂.

The two commutative laws and the associative law of multiplication are obvious. Now:

a/b c/d + a/b e/f = ab (cf + ed)b²df = a/b(c/d + e/f) = (c/d + e/f)a/b

the elements a/b form therefore a commutative ring. 0/b is its zero-element. For a ¹ 0,

For every particular element a of A, there is an element ca/c of Q(A). This element is uniquely determined by a, since

The elements of type (a) form therefore a sub-ring, say A' of Q{A), which is homomorphic to A. If a ¹ b, then ca/c ¹ cb/c, whence the homomorphism is an isomorphism. Finally,

Hence every element of Q(A) is the quotient of two elements of A', whence Q{A) is the quotient field of A, and the theorem holds.

Lemma: Let A be a sub-ring of a ring B and A' be a ring which is isomorphic to A, then there exists a ring B' which is isomorphic to B and which contains A' as a sub-ring.

Proof: Denote the elements of A by a₁, a₂, ··· - in general, by the letter a with an index but without a dash - and the remaining elements of B be by b₁, b₂, ··· with subscripts, but without dashes. Consider now an isomorphism J_a, which maps A onto A', and denote by a_j' the element of A' corresponding to a_j, where j runs over all the subscripts which occur. If a_i - a_j = a_k' and a'_i a'_j=a'_m, then, by the isomorphism, a '_i + a '_j = a_k' and a'_i a'_j = a'_m. Create now new elements b '₁, b '₂, ···, corresponding to the elements b₁, b₂, ··· of B. These new elements, together with the elements of A', form a set B' which is in a (1,1) correspondence to B; naturally, the correspondence can be generated by simply affixing a dash to the notations of the elements of B. Addition and multiplication in B will now be defined as follows: If for any elements r, s, t and m of B, r + s = t and r s =m, then r ' + s ' = t ' and r ' s ' = t '. By this definition, B' is made into a ring and the mapping of B onto B' generated by affixing of a dash becomes an isomorphism J; for the elements of A', this isomorphism is identical to J_a, and addition and multiplication of elements of A' is the same as defined initially. Hence the ring A' is a sub-ring of B'.

In particular, if B is a field, then B' is a field and A' is embedded in B' as a sub-ring. Hence application of the lemma to the proceeding theorem yields

Theorem 1: If A is a commutative ring in which 0 cannot be represented as a product of two elements other than 0, then A is a sub-ring of a field which is isomorphic to Q(A).

Definition: A commutative ring containing a unit element is said to be an integral domain if the product of any two of its elements which both differ from zero differs itself from zero.

Hence: An integral domain can be embedded in a field. The close connection between A and its quotient field Q(A) is stated by

Theorem 2: If A is a sub-ring of F, the meet of all the sub-fields of F which contain A is isomorphic to Q(A).

Proof: As has been shown in 2.2.7, the meet X of those sub-fields consists of the quotients of the elements of A. Addition, subtraction and multiplication of these quotients are given by 2.2.7 (3), (4) and (5). There corresponds to every element of Q(A) an element of X and since the formulae (1) and (2) of 2-28 for rational operations in Q(A) tally with the corresponding formulae of 2.2.7 for the elements of X, the field X is isomorphic to Q(A). Since Q(A) is a field, the theorem follows from the corollary in 2.2.3.

2.2.9.1 Identification: The notion of isomorphism and the Lemma of 2.2.9 above can be generalized. Consider any system of mathematical objects which may be subject to certain operations. Modules and rings are instances of such systems; the objects are called elements. In a module, there is one operation (addition), in a ring there are two operations (addition and multiplication). Another example of a system of this kind is the affine space; its objects are points and vectors; the operations are: Addition of vectors, multiplication of vectors by real numbers, addition of a point and a vector. A system is therefore not uniquely determined only by the objects; naturally, the same set of objects furnishes different systems, if the operations differ. In the same module, multiplication can be introduced possibly in more than one way; thus, one may construct two rings which are different, though composed of the same elements. If in the geometrical system just introduced as affine space, in addition to the operations already introduced, the relation of scalar product is established, the affine space becomes a metric space which is not the same system as the affine space. Let now S and S' be any two systems with the same operations, but possibly different elements, and let there be a (1,1)-correspondence J generated by mapping the elements a, b, ··· of S onto the elements a', b', ··· of S'; moreover, let R be one of the given operations by which a, b, ··· yields R(a_n , b_n, ··· ) = k_n . If R(a '_n , b '_n, ··· ) = k'_n is the element of S' onto which k_n is mapped by J, then R is said to be invariant for J. If every operation of S is invariant, then J is said to be an isomorphism and S is isomorphic to S'. Obviously, the isomorphism satisfies the three conditions for an equivalence. For rings and fields, this general definition of isomorphism tallies with the earlier definitions. The lemma of 2.2.8 can. be generalized now in the following way:

Let S and T be two systems with the same operations and let every element of S be an element of T (i.e., S is a subsystem of T). If S' is isomorphic to S, then there exists a system T' which is isomorphic to T and which contains S' as a sub-system.

This lemma is proved in the same way as the lemma in 2.2.8. The reader should work out the proof as an exercise.

Often, isomorphic systems are considered to be equal - different representations of the same thing. For example, one speaks of the affine plane although there are different planes which are only isomorphic, but for affine planimetry one needs to consider the common properties of all these planes, whence it is convenient to take these planes as representations only of the plane. It is not always possible to proceed in this manner; in Solid Geometry, one has to consider simultanously different (isomorphic) planes, say p₁ and p₂, and these planes may intersect in a line, say s, whereas, if p₁ and p₂are considered to be the same plane, every point of them is a common point; thus one has to distinguish in this case between different isomorphic systems. It is similar in Algebra. If the properties of a particular ring R are investigated, it is not necessary to make any distinction between R and a ring R' which is isomorphic to R. On the other hand, if one discusses a ring S which contains two isomorphic rings R and R', one must make distinction between them. However, there occur later on cases when two isomorphic systems S₁ and S₂ will not be considered as subsystems of a larger system, for example, when by the introduction of S₂, the system S₁ becomes superfluous. In this case, there is no harm to identify them, i.e., to consider them only as the representatives or as different names of one and the same thing.

Consider now the ring A of the last theorem of 2.2.8, then its quotient field Q(A) contains a sub-ring A' and isomorphic to A. It follows from the lemma that A can be embedded in a field B which is isomorphic to Q(A); it is now very convenient to identify B with Q(A) and A with A'. Of course, if on building up elementary arithmetic, one extends the ring of the integral numbers, at first, the quotient field of the fractions a/b is introduced, and then the fractions a/l are identified with the integral numbers a; otherwise it would be necessary to make a distinction between the fractions a/b and the rational numbers a:b. That could be done; one is not loosing any logically important step in renouncing identification, but the mathematical language will become very heavy and overloaded by isomorphisms. In order to understand this clearly, a short review of the steps leading from the notion of natural number to the notion of complex number may be helpful.

1. Out of the two signs +, - and the natural numbers a, form all the pairs + a and - a. These pairs together with a new symbol 0 form a system. Addition and multiplication are defined now in such a manner that the system becomes a ring R and the sub-system of positive numbers + a is isomorphic to the system of the natural numbers. Natural numbers are identified with positive numbers. R is the ring of integral numbers.

2. Form Q(R) and identify R with the sub-ring of the factors with denominator 1. Q(R) is the field of the rational numbers.

(3). Form the Dedekind sections in the ordered system of the rational numbers. For a suitable definition of addition and multiplication, these form a field D. The prime-field P of D consists of those sections which are determined by rational numbers. P is identified with Q(R) and D is called the field of the real numbers.

(4). Form pairs (a, b) of real numbers a, b. For suitably chosen operations of addition and multiplication, these pairs form a field F and the elements (a 0) form a sub-field which is isomorphic to D. Identify D with this sub-field. F is the field of the complex numbers.

Thus four identifications are performed to obtain the complex numbers starting from the natural numbers. It is well known that there are also different ways; for example, one can define real numbers as continued fractions, or decimal fractions, etc. Similarly, complex numbers may be introduced by classes of residues, etc. Obviously, a Dedekind section is a thing which differs from a continued fraction. The real numbers, defined by continued fractions, form a field which by its substance differs from the field of the real numbers defined as Dedekind sections; however, the two fields are isomorphic. Every mathematical statement holding in one of these fields holds also in the other one. In investigations of mathematical logic, it may be necessary to consider these two fields simultanously, and therefore to make a distinction between them. In Pure mathematics there is no reason to do so, whence we are justified in identifying these two fields (and some others) and to speak of the field of the real numbers.

Suppose x is commutative with every element of the ring. From the laws of addition and multiplication (cf. 2.1) as far as they are must be satisfied in a ring of this kind, it follows that

If v is greater than n and m, s_n = d_n = 0, similarly for m > n + m, g_m = 0. Independently of x, the element (1) of the ring is not altered, if some terms with coefficients equal to zero are added. For particular values of x, such, expressions may determine the same element, even if they differ in every coefficient, for example, x— 2 and x² + 2x - 4 are equal for x = 1. It is of fundamental importance that there exists a class of rings in which two elements (1) are equal if and only if corresponding coefficients are equal (terms with zero-coefficients being omitted). Every ring can be extended to a ring of this kind by the operation which will be described now.

2.3.2 Definition of a polynomial: For the above mentioned purpose, one starts from a ring R, the elements of which are denoted by

Introduce now a symbol x which is not used for the notation of elements of R. Then create new elements, polynomials in x over R which are denoted in the same manner in (1) of 2.3.1. It may be emphasised that the polynomials are not yet elements of a ring, but they will become so by suitable definitions. The system of these polynomials is denoted by

As usual, the elements a₀, a₁, ··· , a_n are called coefficients, the symbol x is said to be an indeterminate. Two polynomials are considered to be equal if and only if, after omission of the terms with zero-coefficients, they tally in every coefficient. This definition of equality is admissible, as it obviously satisfies the laws of reflexivity, symmetry and transitivity. As abbreviation, we are allowed to omit terms with zero-coefficients; moreover, we may omit any coefficient which is equal to 1 (provided that a unit element 1 exists in R). Thus the symbol x can itself be considered to be a polynomial

This formula is not trivial, as - up to now - the sign + in 2.3.1 (1) is a mere symbol and not the sign of addition. Addition, multiplication and subtraction of polynomials have not yet been defined; suitable definitions follow. ??

Definition: If a_i = 0 for i > d, but a_d¹ 0, then d is called the degree of the polynomial S a_ixⁱ. If every coefficient is zero, the degree is equal to — 1.

By this definition, there is allotted to every polynomial a definite integral number ³ -1 as its degree. Equal polynomials have the same degree. The polynomials of degree - 1 are all equal, whereas there exist for degrees ³ 0 different polynomials of the same degree. The polynomials of degree < 1 are of the type

of R[x]. By the definition of equality of polynomials, every polynomial (3) is equal to a polynomial a₀. Thus there is a (1,1)-correspondence between the elements of the ring R and those of R⁰[x], so that corresponding elements are written in the same way. The distinction between R and R⁰[x] will disappear later on.

Definition: The sum [product] of two polynomials of R[x] is a polynomial R[x]; the coefficients are determined by (2') [by (4')] of 2.3.1.

Theorem: R[x] is a ring. R⁰[x] is a sub-ring of R[x] which is isomorphic to R, corresponding elements being denoted in the same way. A polynomial is the sum of single-term polynomials a₀, a₁x, ··· , a_nxⁿand each polynomial a_nxⁿ is the product of the polynomial a_n (of degree < 1) and the polynomial xⁿ.

Proof: In order to show that R[x] is a ring, one has to prove that the commutative law of addition, the associative laws of addition and multiplication, the distributive laws and the law of inverse-existence for addition apply. Since these laws hold in R, one can easily derive them for R[x] with the aid of the formulae of 2.3.1. The commutative law of addition follows directly from (2) and (2') of 2.3.1

i.e., the associative law for the addition of polynomials, Similarly,one shows that

are both equal to S u_n xⁿ , where u_n . It may be left to the reader to check the two distributive laws in the same manner! lf d_n is defined by 2.3.1 (3'), Sd_n xⁿ + S b_nxⁿ = S a_nxⁿ follows from (2') of 2.3.1. Hence R[x] is a ring. The remaining propositions of the theorem are immediate consequences of the definition of addition and multiplication of polynomials.

Since R⁰[x] is isomorphic to the ring R, one identifies the elements of R⁰[x] with the elements of R which are already denoted in the same manner. It is therefore not no longer necessary to distinguish between the polynomial a₀ (of degree < 1) and the element a₀ of R. The ring R[x] is an extension of the ring R, since by the identification carried out just before R becomes a sub-ring of R[x]. Since every sub-ring (and even every sub-module) of a ring (a module) contains the null-element of the ring (module), the rings R[x] and R have the same null-element, that is, the null-element 0 of R is also the null-element of R[x]. Moreover, if R contains a unit element 1, this unit element is also the unit element of R[x].

Since a polynomial has been proved to be the sum of its terms (which are polynomials with at most one coefficient different from zero), one can interchange the terms without changing the polynomial; for example, one can write the terms in the opposite order

Exercises: Let R be a ring which may or may not be commutative, but contains a unit element:

1. Prove that xⁿ is commutative with every polynomial of R[x].
2. Let R[x] be a sub-ring of a ring R₁; prove that R[x] is the meet of all those sub-rings of R which contain R and x.

Proof: By the theorem of 2.3.3, it has been shown that R[x] is a ring and only the commutativity of the multiplication has to be proved. Of course, the commutativity follows from 2.3.1 (4') and

'Proof: As a consequence of Theorem 1, R[x] is a commutative ring which contains the unit element 1; it need only be proved that if

are of degree ³ 0, then the same is true for the product

Without loss of generality, we can assume that a_n ¹ 0 and b_m¹ 0, whence g_n+m = a_nb_m ¹ 0. Hence the theorem has been proved.

Theorem 3. If R is a sub-ring of a field, then the degree of the product of two polynomials of R[x] is the sum of the degrees of the factors, provided the factors are both of degree ³ 0.

Exercise: Show that the condition R is an integral domain cannot be replaced in Theorem 3 by R is a commutative ring.

Proof: If R consists only of the null-element, then R[x] is equal to R and, as it consists of one element only, it is not a field. Let a ¹ 0 be an element of R, then R[x] contains a x. If R[x] is a field, R is a sub-ring of a field; hence Theorem 3 can be applied. On the other hand, there exists in R[x] a polynomial of degree, say m, which is inverse to ax. The product of the two polynomials is equal to 1, whence it is a polynomial of degree 0. Now m > -1, but, by Theorem 3, one has m + 1 = 0, which is a contradiction.

2.3.5 Integral functions: Let C be a commutative ring containing the unit element 1; then the same holds for C[x]. Let a₀, a₁ , ··· , a_n be elements of C and l an arbitrary element of C[x], then

is again an element of C[x]. The correspondence mapping l on f(l) is called an integral rational function over C or briefly (as we are only concerned with Algebra) an integral function over C. These functions map the elements of C onto elements of C, and the elements of C[x] onto elements of C[x]. In particular, 0 is mapped onto a₀and x (which indeed is an element of C) as 1 is supposed to be an element of C is mapped onto the polynomial f(x) which has the same coefficients as (1). On the other hand, given any polynomial in x over C, there exists an integral function over C with the same coefficients; one need only replace the indeterminate x by the variable l running over the elements of C[x] or a subset of them.

and any particular element of C[x], say c. Let there correspond to every element f(x) of C the element f(l) which also belongs to C[x]. Let f₁(x) and f₂(x) be any two elements of C[x]

Then the coefficients a_i, b_j, s_n , g_m are interconnected by 2.3.1 (2') and (4').

Since C[x] is a commutative ring, l is commutative with the elements a_i and b_j, hence (cf. 2.3.1) the addition and the multiplication of S a_il^ij and S b_jl^j is done also in accordance with 2.3.1 (2), (2'), (4) and (4'). Hence

The correspondence f(x) ® f(l) is therefore a homomorphism. It maps the ring of all the elements of C[x] onto a sub-ring C, which is homomorphic to C[x]

Let l = b be an element of C. Then every element of C_b is an element of C; on the other hand, the polynomials x + (a— b) of C[x] are mapped onto a, which is supposed to be an arbitrary element of C. Hence C_b = C. Thus every element of C generates an homomorphism which maps C[x] onto C.

By a homomorphism, the null-element is always mapped onto the null-element. If therefore

Theorem: Every equation between elements of C[x] remains correct, if one puts for x any particular element of C[x].

Note that the commutativity of C[x] is essential for the validity of this theorem.

are also rings. T[y] consists of the polynomials in y of which the coefficients are polynomials in x over R, i.e., it consists of the sums

This statement is readily generalized for any number of indeterminates. We can now extend R to

without making any distinction regarding the order in which the extension is performed.

where j₁(x) is a polynomial in x over the same ring as j(x), provided this ring contains a unit element.

Let D be an integral domain; as in 2.2.5, the sum of n terms, each being equal to 1, is denoted by n. Now let f(x) be a polynomial of D(x) = T, then f(x + y) is a polynomial of T[y]:

where f '(x) is a polynomial in x over D and is uniquely determined by f(x). The polynomial f'(x) is called the derivative of f(x). If D consists of real numbers only, this notation tallies with what in Analysis is called the derivative of an integral rational function. The reader knows that in Analysis the notion of derivative applies to a much larger class of functions of a real variable than only rational functions. Here, a derivative is defined for polynomials only, but the coefficients are not necessarily real numbers. The formulae for the derivatives of sums and products are the same as in Analysis and even the proofs are nearly the same, the only consideration being that there is no passage to a limit. Readers are advised to compare carefully the following proofs with those given in analysis in order to gain a clear understanding of the difference between an indeterminate and a variable which takes real values some of which possibly make the functions meaningless.

Since D and therefore D[y] are integral domains and y is not the null element of D[y], the factor in brackets is zero. After setting y = 0, (3) yields

2.3.7 Homogeneous polynomials: Again let D be an integral domain, x₁, x₂, ··· , x_n indeterminates and introduce the notation

Every polynomial f(x₁, ··· , x_n) of S can be considered to be a polynomial in x over D

is a necessary condition for f(x₁, ··· , x_n) to be homogeneous of degree m. Obviously, the condition is also sufficient. Forming the n derivatives and multiplying each of them by the corresponding indefinite x_k, one finds

where, as in 2.2.6 (3)) s_j stands for the element which is obtained by taking the unit element s_j times. From (6) follows by addition