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Abstract algebra
Abstract algebra is the field of mathematics concerned with the study of algebraic structures such as groups, rings and fields. The term abstract algebra is used to distinguish the field from "elementary algebra" or "high school algebra", which teach the correct rules for manipulating formulas and algebraic expressions involving real and complex numbers, and unknowns. Abstract algebra was at times in the first half of the twentieth century known as modern algebra. The term abstract algebra is sometimes used in universal algebra where most authors use simply the term "algebra". In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. For example, the set of integers is a group under the operation of addition. The branch of mathematics which studies groups is called group theory. The historical origin of group theory goes back to the works of Evariste Galois ( 1830 ), concerning the problem of when an algebraic equation is soluble by radicals. Previous to this work, groups were mainly studied concretely, in the form of permutations; some aspects of abelian group theory were known in the theory of quadratic forms.
A great many of the objects investigated in mathematics turn out to be groups. These include familiar number systems, such as the integers, the rational numbers, the real numbers, and the complex numbers under addition, as well as the non-zero rationals, reals, and complex numbers, under multiplication. Another important example is given by non- singular matrices under multiplication, and more generally, invertible functions under composition. Group theory allows for the properties of these systems and many others to be investigated in a more general setting, and its results are widely applicable. Group theory is also a rich source of theorems in its own right. Groups underlie many other algebraic structures such as fields and vector spaces. They are also important tools for studying symmetry in all its forms; the principle that the symmetries of any object form a group is foundational for much mathematics. For these reasons, group theory is an important area in modern mathematics, and also one with many applications to mathematical physics (for example, in particle physics).
The above axioms are not strictly minimal from a logical viewpoint; they contain a small amount of redundancy. However, the difference is slight and in practice one usually just checks the above axioms. It should be noted that there is no requirement that the group operation be commutative, that is there may exist elements such that a * b ≠ b * a. A group G is said to be abelian (after the mathematician Niels Abel) (or commutative ) if for every a , b in G , a * b = b * a. Groups lacking this property are called non-abelian. The order of a group G , denoted by | G | or o( G ), is the number of elements of the set G. A group is called finite if it has finitely many elements, that is if the set G is a finite set. Note that we often refer to the group ( G , * ) as simply " G ", leaving the operation * unmentioned. But to be perfectly precise, different operations on the same set define different groups.
Notation for groups
Usually the operation, whatever it really is, is thought of as an analogue of multiplication, and the group operations are therefore written multiplicatively. That is: We write " a · b " or even " ab " for a * b and call it the product of a and b ; We write "1" for the identity element and call it the unit element ; We write " a − " for the inverse of a and call it the reciprocal of a. However, sometimes the group operation is thought of as analogous to addition and written additively : We write " a + b " for a * b and call it the sum of a and b ; We write "0" for the identity element and call it the zero element ; We write "− a " for the inverse of a and call it the opposite of a. Usually, only abelian groups are written additively, although abelian groups may also be written multiplicatively. When being noncommittal, one can use the notation (with "*") and terminology that was introduced in the definition, using the notation a − for the inverse of a. If S is a subset of G and x an element of G , then, in multiplicative notation, xS is the set of all products { xs : s in S }; similarly the notation Sx = { sx : s in S }; and for two subsets S and T of G , we write ST for { st : s in S , t in T }. In additive notation, we write x + S , S + x , and S
- T for the respective sets.
Not a group: the integers under multiplication On the other hand, if we consider the operation of multiplication, denoted by "·", then ( Z ,·) is not a group: If a and b are integers then a · b is an integer. (Closure) If a , b , and c are integers, then ( a · b ) · c = a · ( b · c ). (Associativity) 1 is an integer and for any integer a , 1 · a = a · 1 = a. (Identity element) However, it is not true that whenever a is an integer, there is an integer b such that ab = ba = 1. For example, a = 2 is a integer, but the only solution to the equation ab = 1 in this case is b = 1/2. We cannot choose b = 1/2 because 1/2 is not an integer. (Inverse element fails ) Since not every element of ( Z ,·) has an inverse, ( Z ,·) is not a group. The most we can say is that it is a commutative monoid.
An abelian group: the nonzero rational numbers under multiplication Consider the set of rational numbers Q , that is the set of numbers a / b such that a and b are integers and b is nonzero, and the operation multiplication, denoted by "·". Since the rational number 0 does not have a multiplicative inverse, ( Q ,·), like ( Z ,·), is not a group. However, if we instead use the set Q \ {0} instead of Q , that is include every rational number except zero, then ( Q \ {0},·) does form an abelian group (written multiplicatively). The inverse of a / b is b / a , and the other group axioms are simple to check. We don't lose closure by removing zero, because the product of two nonzero rationals is never zero. Just as the integers form a ring, so the rational numbers form the algebraic structure of a field. In fact, the nonzero elements of any given field form a group under multiplication, called the multiplicative group of the field.
Ring (mathematics)
In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have similar (but not identical) properties to those familiar from the integers. The branch of abstract algebra which studies rings is called ring theory. Formal definition A ring is a set R equipped with two binary operations + and ·, called addition and multiplication , such that: ( R , +) is an abelian group with identity element 0: o ( a + b ) + c = a + ( b + c ) o a + b = b + a o 0 + a = a + 0 = a o ∀ a ∃(− a ) such that a + − a = − a + a = 0 ( R , ·) is a monoid with identity element 1: o 1· a = a ·1 = a o ( a · b )· c = a ·( b · c ) Multiplication distributes over addition: o a ·( b + c ) = ( a · b ) + ( a · c ) o ( a + b )· c = ( a · c ) + ( b · c )
As with groups the symbol · is usually omitted and multiplication is just denoted by juxtaposition. Also the standard order of operation rules are used, so that e.g. a + bc is an abbreviation for a +( b · c ). Although ring addition is commutative (i.e. a + b = b + a ), note that the commutativity for multiplication ( a · b = b · a ) is not among the ring axioms listed above. Rings that also satisfy commutativity for multiplication (such as the ring of integers) are called commutative rings. Not all rings are commutative. Also note that an element of a ring need not have a multiplicative inverse. An element a in a ring is called a unit if it is invertible with respect to multiplication, i.e., if there is an element b in the ring such that a · b = b · a = 1. If that is the case, then b is uniquely determined by a and we write a − = b. The set of all units in R forms a group under ring multiplication; this group is denoted by U ( R ).
Simple theorems From the axioms, one can immediately deduce that, for all elements a and b of a ring, we have 0 a = a 0 = 0 (−1) a = − a (− a ) b = a (− b ) = −( ab ) ( ab ) − = b − a − if both a and b are invertible
Constructing new rings from given ones
If a subset S of a ring R is itself a ring with the same operations (restricted to S ), and the identity element 1 of R is contained in S , then S is called a subring of R. The center of a ring R is the set of elements of R that commute with every element of R ; that is, c lies in the center if cr = rc for every r in R. The center is a subring of R. We say that a subring S of R is central if it is a subring of the center of R. The direct sum of two rings R and S is the cartesian product R × S together with the operations ( r 1 , s 1 ) + ( r 2 , s 2 ) = ( r 1 + r 2 , s 1 + s 2 ) and ( r 1 , s 1 )( r 2 , s 2 ) = ( r 1 r 2 , s 1 s 2 ). Given a ring R and an ideal I of R , the quotient ring (or factor ring ) R / I is the set of cosets of I together with the operations ( a+I ) + ( b+I ) = ( a + b ) + I and ( a+I )( b+I ) = ( ab ) + I. Since any ring is both a left and right module over itself, it is possible to construct the tensor product of R over a ring S with another ring T to get another ring provided S is a central subring of R and T.
Integral domain
Every field is an integral domain. Conversely, every Artinian integral domain is a field. In particular, the only finite integral domains are the finite fields. Rings of polynomials are integral domains if the coefficients come from an integral domain. For instance, the ring Z [X] of all polynomials in one variable with integer coefficients is an integral domain; so is the ring R [X,Y] of all polynomials in two variables with real coefficients. The set of all real numbers of the form a + b √2 with a and b integers is a subring of R and hence an integral domain. A similar example is given by the complex numbers of the form a + bi with a and b integers (the Gaussian integers ). The p-adic integers. If U is a connected open subset of the complex number plane C , then the ring H( U ) consisting of all holomorphic functions f : U -> C is an integral domain. The same is true for rings of analytical functions on connected open subsets of analytical manifolds. If R is a commutative ring and P is an ideal in R , then the factor ring R/P is an integral domain if and only if P is a prime ideal.
Divisibility, prime and irreducible elements
If a and b are elements of the integral domain R , we say that a divides b or a is a divisor of b or b is a multiple of a if and only if there exists an element x in R such that ax = b. If a divides b and b divides c , then a divides c. If a divides b , then a divides every multiple of b. If a divides two elements, then a also divides their sum and difference. The elements which divide 1 are called the units of R ; these are precisely the invertible elements in R. Units divide all other elements. If a divides b and b divides a , then we say a and b are associated elements. a and b are associated if and only if there exists a unit u such that au = b. If q is a non-unit, we say that q is an irreducible element if q cannot be written as a product of two non-units. If p is a non-zero non-unit, we say that p is a prime element if, whenever p divides a product ab , then p divides a or b. This generalizes the ordinary definition of prime number in the ring Z , except that it allows for negative prime elements. If p is a prime element, then the principal ideal ( p ) generated by p is a prime ideal. Every prime element is irreducible (here, for the first time, we need R to be an integral domain), but the converse is not true in all integral domains (it is true in unique factorization domains, however).
Field (mathematics)
In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers.
Introduction
Fields are important objects of study in algebra since they provide a useful generalization of many number systems, such as the rational numbers, real numbers, and complex numbers. In particular, the usual rules of associativity, commutativity and distributivity hold. Fields also appear in many other areas of mathematics; see the examples below. When abstract algebra was first being developed, the definition of a field usually did not include commutativity of multiplication, and what we today call a field would have been called either a commutative field or a rational domain. In contemporary usage, a field is always commutative. A structure which satisfies all the properties of a field except for commutativity, is today called a division ring or sometimes a skew field , but also non- commutative field is still widely used. The concept of a field is of use, for example, in defining vectors and matrices, two structures in linear algebra whose components can be elements of an arbitrary field. Galois theory studies the symmetry of equations by investigating the ways in which fields can be contained in each other. See field theory for more.
Definition
A field is a commutative ring ( F , +, *) such that 0 does not equal 1 and all elements of F except 0 have a multiplicative inverse. Spelled out, this means that the following hold: Closure of F under + and * For all a , b belonging to F , both a + b and a * b belong to F (or more formally, + and * are binary operations on F ). Both + and * are associative For all a , b , c in F , a + ( b + c ) = ( a + b ) + c and a * ( b * c ) = ( a * b ) * c. Both + and * are commutative For all a , b belonging to F , a + b = b + a and a * b = b * a. The operation * is distributive over the operation + For all a , b , c , belonging to F , a * ( b + c ) = ( a * b ) + ( a * c ). Existence of an additive identity There exists an element 0 in F , such that for all a belonging to F , a + 0 = a. Existence of a multiplicative identity There exists an element 1 in F different from 0, such that for all a belonging to F , a * 1 = a. Existence of additive inverses For every a belonging to F , there exists an element − a in F , such that a + (− a ) = 0. Existence of multiplicative inverses For every a ≠ 0 belonging to F , there exists an element a − in F , such that a * a − = 1.
