Algebra

[dài shù]

Branch of mathematics

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Algebra , is the number of studies number The branch of mathematics that deals with general solutions and properties of algebraic equations (systems). elementary algebra It is generally taught in middle school to introduce the basic idea of algebra: study what happens when we add or multiply numbers, and understand the concept of variables and how to establish them polynomial And find their roots. Algebra studies not only numbers, but various abstract structures. We only care about various relationships and their properties, but we don't care about the question of "what is number itself". The common types of algebraic structures are groups ring , domain, module linear space Etc.^[1]

Chinese name: Algebra
Foreign name: algebra
Discipline: mathematics
Discipline characteristics: abstract

Important theories: Galois theory
Common types: Symmetric algebra, tensor algebra
Interpretation: A branch of mathematics that studies numbers, quantities, relationships, structures, and general solutions and properties of algebraic equations (systems)

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six The Origin of Words
seven Algebra of commutative rings

introduce

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In ancient times, when arithmetic In order to find systematic and more general methods to solve various quantitative problems Quantitative relation The problem of algebraic equation The principle of elementary algebra 。

Algebra is evolved from arithmetic, which is beyond doubt. As for when it came into being Algebra It is difficult to explain this subject clearly. For example, if you think "algebra" refers to the skill of solving algebraic equations represented by symbols such as bx+k=0. This "algebra" was developed only in the sixteenth century.

definition

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Algebra is a branch of mathematics. Traditional algebra uses expressions with characters (variables) for arithmetic operations, and characters represent unknowns or undetermined numbers. If division is not included (except for integer division), each expression is a polynomial with rational coefficients. For example: 1/2 xy+1/4z-3x+2/3 An algebraic equation (see Equation) expresses the conditions imposed on variables by making polynomials equal to zero. If there is only one variable, then the equation will be satisfied by a certain number of real or complex numbers - its root. An algebraic number is the root of an equation.

Theory of Algebraic Numbers—— Galois theory It is one of the most satisfying branches of mathematics. Building this theory Galois (Evariste Galois, 1811-1832) died in a duel at the age of 21. He proved that there could be no algebraic formula to solve the quintic equation. In his method, he also proved that some famous geometric problems cannot be solved with a ruler and a compass (the cube is doubled, and an angle is bisected). The theory of algebraic equations with more than one variable belongs to algebraic geometry. Abstract algebra deals with generalized mathematical structures. They are similar to arithmetic operations. See, for example: Boolean algebra (BOOLEAN ALGEBRA); group (GRO-UPS); matrix (MATRICES); Quaternion (QUA-TERNIONS ); vector (VECTORS)。 These structures are characterized by axioms (see axiomatic method AXIOMATICMETHOD). Of particular importance are the associative law and the commutative law. Algebraic method simplifies the solution of the problem to the operation of symbolic expression, which has infiltrated into various branches of mathematics.

Let K be an commutator The vector space E on K is called algebra on K, or K-algebra. If the bilinear mapping from E × E to E is assigned, in other words, the algebraic structure of set E defined by the following three given rules is assigned:

——It is recorded as the composition rule of addition (x, y) ↦ x+y;

——It is recorded as the second composition rule of multiplication (x, y) ↦ xy;

——The mapping from K × E to E (α, x) ↦ α x recorded as multiplication is a rule of action;

These three principles meet the following conditions:

a) Given the first and third rules, E is a vector space on K;

b) For any triplet (x, y, z) of the element of E

x(y+z)=xy+xz(y+z)x=yx+zx;

c) For any element pair (α, β) of K and any element pair (x, y) of E, there is (α x) (β y)=(α β) (xy)

Let A be a non empty set Give the set of all mappings from A to K ℱ (A, K) the following three rules:

Then ℱ (A, K) is an algebra on K, which is naturally called the mapping algebra from A to K. When A=N, algebra ℱ (A, K) is called the element sequence algebra of K

Whether in algebra or in analysis, algebraic structure is one of the most common structures. At the end of the first half of the 19th century, with the establishment of Hamilton's quaternion theory, the study of noncommutative algebra has begun In the second half of the nineteenth century, with M S. Li's work on non associative algebra appeared By the beginning of the twentieth century, algebra had been greatly expanded due to the abandonment of the restriction of real number field or complex number field as an operator field

And exterior algebra, Symmetric algebra , tensor algebra, Clifford algebra Together, algebraic structure is also established in multilinear algebra

Traceability

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Diophantus, an ancient Greek mathematician

If we don't require algebraic symbols to be as concise as they are now, then, Algebra The emergence of "can be traced back to an earlier era.".

Westerners will^{[7 ]} Greek mathematicians Difantu As the originator of algebra, it is ancient times who really founded algebra Arab Empire Mohammed Ibn Moussa, the great mathematician of the period (known as "Khwarazmi" in China, born and died about 780-850 AD). In China, algebraic problems expressed in words appeared earlier.

"Algebra", as a proper noun of mathematics and representing a branch of mathematics, was officially used in China as early as 1859. That year, mathematicians in the Qing Dynasty Li Shanlan I translated a book written by the British Demegan with the Englishman Wileyali. The name of the translation is Algebra. Of course, the contents and methods of algebra have long been produced in ancient China, such as《 Chapter Nine Arithmetic 》In Equation problem 。

The origin of algebra can be traced back to ancient times Babylon Era of [1] At that time, people developed more advanced Arithmetic system , so that it can be calculated in an algebraic way. After being systematically used, they can list equations containing unknowns and solve them. These problems are generally used today linear equation 、 Quadratic equation and Indefinite linear equation And so on. In contrast, most of the Egyptian And the Western Yuan Dynasty First century Most mathematicians in India, Greece and China usually solve such problems by geometric methods, such as in Rhind Mathematical Papyrus 、 Rope meridian 、 Geometric primitives and Chapter Nine Arithmetic And so on. Greece's work in geometry, taking geometry as its classic, provided a framework for generalizing the formula for solving specific problems into a more general system for describing and solving algebraic equations.

Algebra originates from the Arabic word "al jabr", which comes from al-Kitāb al-muḫtaṣar fī ḥisāb al-ğabr wa-l-muqābala The title of this book refers to the abstract of the calculation of the transfer and merger of similar terms, which is Persian Muslim mathematician Hua La Zi Mi It was written in 820. Al Jabr means "reunion". Traditionally, Greek mathematicians Difantu Known as the "father of algebra", his achievements are still useful today, and he gave a detailed explanation for solving quadratic equations. While those who support Diophantine argue that the algebra in Al Jabr is more basic than that in Arithmetic, and Arithmetic is simplified while Al Jabr is completely literal. [3] Another Persian mathematician Omar Khayyam Developed Algebraic geometry And found Cubic equation The general geometric solution of. Indian mathematician Mahavira and Boshgara And Chinese mathematicians Zhu Shijie Solved a lot of three, four Five times And higher polynomial The solution of the equation is obtained.

Another key event for further development of algebra is the general algebraic solution of cubic and quartic equations, which was developed in the middle of the 16th century. determinant The concept of Japanese mathematician Guan Xiaohe And ten years later Leibniz It continues to develop in order to matrix To solve Linear equations The answer comes from. Gabriel Kramer The same work was done on matrices and determinants in the 18th century. Abstract algebra The development of Galois theory and Regular number On the issue of.

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elementary algebra

Elementary algebraic operations

Elementary algebra studies numbers and characters Algebraic operation The theory and method, more specifically, is to study the sum of real numbers and complex , as well as the mathematical branch of algebraic operation theory and method of algebraic expressions with them as coefficients subject 。

Explain what elementary algebra is in popular language, that is, if we define arithmetic as studying the characteristics of apples, pears, oranges, grapes, etc. respectively, then elementary algebra is studying the commonalities of fruits.

The basic contents of elementary algebra are as follows

Three kinds of numbers rational number Irrational number 、 complex ；

Three types—— Integral form 、 fraction 、 Radical (collectively referred to as algebraic expression);

Three kinds of equations—— Integral equation 、 Fractional equation 、 Irrational equation (collectively referred to as algebraic equation), and algebra formed by a finite number of algebraic equations Equations 。

It is worth noting that : According to the definition of the equation, as long as it contains unknown number Of equation That's the equation. The reason why we should emphasize " algebraic equation ", because in addition to algebraic equations, there are Transcendental equation (i.e. non algebraic elementary equations, including Exponential equation 、 Logarithmic equation 、 Trigonometric equation , inverse trigonometric equation, etc.) differential equation 、 Difference equation 、 integral equation And many other forms of equations. The latter categories obviously do not belong to algebra. Some contents about the history of mathematics often define algebra as " solve equations As the core discipline, it is mainly because the knowledge about algebraic equations in history has been applied in calculus, etc Modern mathematics The branch was studied long before it was established. Since there were no Calculus How can mathematicians think of establishing differential equation What about the concept of?

elementary algebra The content of (elementary algebra) is basically equivalent to the content of algebra courses set up in current middle schools, but it is not completely the same. For example, strictly speaking, the concept, arrangement and combination of numbers should be included in the content of arithmetic; Function is the content of analytical mathematics; Inequality The solution of solve equations However, inequality, as a method of estimating values, essentially belongs to Analytical mathematics The scope of; Coordinate method is used to study analytic geometry, etc. These are just a kind of arrangement method formed in history.

Elementary algebra is the continuation and promotion of arithmetic Algebraic expression And solving algebraic equations. Algebraic operation It is characterized by adding, subtracting, multiplying, dividing, and square root extraction only for a limited number of times. All elementary algebra has ten rules. This is the key point for learning elementary algebra.

rule

Five Basics Operational law ： Additive commutative law 、 Associative law of addition 、 Commutative law of multiplication 、 Associative law of multiplication 、 Distributive law ；^[2]

Basic properties of two equations: add (subtract) a number on both sides of the equation at the same time, and the equation remains unchanged; Both sides of the equation are multiplied (divided) by a non-zero number at the same time, and the equation remains unchanged;

Three exponential laws: Power of same base Multiplication, base number unchanged index Addition; Power Power , base constant index multiplication; The power of the product is equal to the product of the power.

Elementary algebra further develops in two aspects: on the one hand, it studies the system of linear equations with more unknowns; On the other hand, the research on unknowns is more frequent (unitary) Higher order equation 。 At this time, algebra has moved from elementary algebra to Advanced Algebra The direction of linear algebra And (unary) Polynomial algebra "Two major sectors.

1 ° a-b=0, if and only if a=b;

2 ° a+b=0, if and only if a=- b, or b=- a;

3 ° a · b=0, if and only if a=0, or b=0;

4° (a-b) two =0, if and only if a=b.

Advanced Algebra

Research object

Advanced algebra is Algebra The general term for the advanced stage, which includes many branches. Higher algebra offered in universities generally includes two parts: linear algebra , polynomial algebra.

Advanced algebra further expands its research objects on the basis of elementary algebra, introducing many new concepts and quantities that are very different from those usually used, such as the most basic ones are sets vector and vector space Etc. These quantities have the operation characteristics similar to numbers, but the research methods and operation methods are more complex. Set is the whole of things with certain attributes; A vector is a quantity that has direction as well as numerical value; Vector space, also called linear space, is a set of many vectors that conform to the rules of certain operations. The object of operation in vector space is no longer just a number, but a vector, and its operation properties are also very different.

Difference and connection with linear algebra

Many people Advanced Algebra and linear algebra Confused, do not understand the difference.^[3]

Higher algebra is a specialized course for college mathematics majors, while linear algebra is a course for science, engineering and some medical majors except mathematics majors.

Solving algebraic equations

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Complex operation

The central content of elementary algebra is solution algebraic equation Therefore, algebra has long been understood as the science of algebraic equations, and mathematicians have also focused on the research of algebraic equations. Its research method is highly computational.^[4]

To discuss algebraic equations, one of the first problems encountered is how to compose the quantitative relations in practice Algebraic expression , and then according to Equimetric relation List the equations. So an important content of elementary algebra is algebraic expression. Due to the different quantitative relations in things, elementary algebra has formed Integral form 、 fraction and Radical These three kinds of algebraic expressions. Algebraic expressions are the incarnation of numbers, so in algebra, they can be Four arithmetic operations , obey Basic operation Law, but also can be carried out Power (This is only limited to the exponential power of rational number) and square root. These six operations are usually called Algebraic operation , to distinguish it from the Arithmetic operation 。

In the process of the generation and development of elementary algebra algebraic equation The research of, also promoted the further development of the concept of number, extended the concept of integer and fraction discussed in arithmetic to the range of rational number, so that numbers include positive and negative integers, positive and negative fractions and zero. This is another important content of elementary algebra, which is the extension of the concept of number.

With rational numbers, the problems that elementary algebra can solve are greatly expanded. However, some univariate polynomial equations still have no solution in the range of rational numbers. Thus, the concept of number was expanded to real number , and further expanded to complex 。

Is there still an algebraic equation that has not been solved in the complex number range, and the complex number must be expanded again? Mathematicians said: No, no. This is a famous theorem in algebra—— Fundamental theorem of algebra 。 This theorem is simply n Sub equation has n Root (s). December 15, 1742 Swiss mathematician Euler He made a clear statement in a letter, and later another mathematician, the German Gaussian A strict proof was given in 1799.

The Origin of Words

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English name of algebra algebra The name comes from the important works of Arab mathematician Hua Lazimi in the 9th century. The work is called "ilm al jabr wa'1 muqabalah", which originally means "the science of reduction and cancellation". When the book was spread to Europe, it was translated as algebra 。 At the beginning of the Qing Dynasty, two volumes of algebra books without authors were introduced into China, which were translated into Algebra and later into Algebra.

Algebra of commutative rings

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In mathematics, an algebra or multivariate ring over a commutative ring is a algebraic structure When the context is not confused, it is usually called algebra.

definition

set up R Is a commutative ring, R On Algebra (or called A -Algebra ）Is the following structure:

aggregate A It's a R -Die.
A There is a binary operation * on, and * is bilinear, namely:

R (a * b)=(ra) * b=a * (rb)

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The most frequently considered case is that R is a domain, which is called Field algebra Some authors also define algebra as algebra over a field.

if A If the multiplication on satisfies the commutativity ab=ba, it is called Commutative algebra ； if A Multiplication on satisfies Associative law A (bc)=(ab) c, it is called "associative algebra", see Associative algebra Entries. Commutative algebra The algebra considered in the course of learning belongs to commutative associative algebra.^[6]

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