why abstract algebra? (outline of chapter 1 of pinter’s...

Why Abstract Algebra? (Outline of Chapter 1 ofPinter’s Book)

Sean Ellermeyer

Kennesaw State University

August 17, 2015

Sean Ellermeyer (Kennesaw State University)Why Abstract Algebra? (Outline of Chapter 1 of Pinter’s Book)August 17, 2015 1 / 15

The word “algebra” comes from “al jabr” in Arabic. The rough translationof this word is “reunion”. This refers to the collecting of like terms tosolve equations. The word was first coined by the mathematician AbuJa’far Muhammad ibn Musa Al-Khwarizmi (790 – 850). Omar Khayyam(1048–1131) defined “algebra” to be “the science of solving equations”.


Linear and Quadratic Equations

Procedures (formulas) for solving linear equations

ax + b = 0, a 6= 0

and quadratic equations

ax2 + bx + c = 0, a 6= 0

were known since ancient times (for instance, in ancient Greek society) butno such procedure was known for solving cubic equations

ax3 + bx2 + cx + d = 0, a 6= 0

or polynomial equations of any degree n > 3. To find such procedures wasa problem that occupied mathematicians for many centuries.


Example of a Cubic Equation

An example of a cubic equation is

4x3 + 12x2 + x + 3 = 0.

The solutions of this equation are x = −3, x = 12 i and x = −12 i (where i

is the complex number defined such that i2 = −1). It is easily checkedthat these numbers are solutions but how do we find them? Is there aformula (along the same lines as the quadratic formula) that we can use tosolve equations of the form

ax3 + bx2 + cx + d = 0, a 6= 0?


Developments During the Italian Renaissance

Girolamo Cardano (1501–1576). To quote from Charles Pinter,Cardano was a physician, an astrologer, a mathematician, acompulsive gambler, a scoundrel and a heretic.

He was initially not successful with his medical practice and he and hiswife had to seek refuge in a poorhouse.

He was eventually able to land a position as a lecturer of mathematics.Later he was very successful in medicine and was the first physician togive a clinical description of typhus fever.Due to his interest in gambling, he took a great interest in probabilityand became an expert on the subject. He wrote the book Book onGames of Chance.He also wrote the famous Ars Magna which means “The Great Art”. Itwas a book that compiled all that was known about algebra at the time.




He was initially not successful with his medical practice and he and hiswife had to seek refuge in a poorhouse.He was eventually able to land a position as a lecturer of mathematics.

Later he was very successful in medicine and was the first physician togive a clinical description of typhus fever.Due to his interest in gambling, he took a great interest in probabilityand became an expert on the subject. He wrote the book Book onGames of Chance.He also wrote the famous Ars Magna which means “The Great Art”. Itwas a book that compiled all that was known about algebra at the time.




He was initially not successful with his medical practice and he and hiswife had to seek refuge in a poorhouse.He was eventually able to land a position as a lecturer of mathematics.Later he was very successful in medicine and was the first physician togive a clinical description of typhus fever.

Due to his interest in gambling, he took a great interest in probabilityand became an expert on the subject. He wrote the book Book onGames of Chance.He also wrote the famous Ars Magna which means “The Great Art”. Itwas a book that compiled all that was known about algebra at the time.




He was initially not successful with his medical practice and he and hiswife had to seek refuge in a poorhouse.He was eventually able to land a position as a lecturer of mathematics.Later he was very successful in medicine and was the first physician togive a clinical description of typhus fever.Due to his interest in gambling, he took a great interest in probabilityand became an expert on the subject. He wrote the book Book onGames of Chance.

He also wrote the famous Ars Magna which means “The Great Art”. Itwas a book that compiled all that was known about algebra at the time.




He was initially not successful with his medical practice and he and hiswife had to seek refuge in a poorhouse.He was eventually able to land a position as a lecturer of mathematics.Later he was very successful in medicine and was the first physician togive a clinical description of typhus fever.Due to his interest in gambling, he took a great interest in probabilityand became an expert on the subject. He wrote the book Book onGames of Chance.He also wrote the famous Ars Magna which means “The Great Art”. Itwas a book that compiled all that was known about algebra at the time.


Niccolò Fontana Tartaglia (1500–1557)

He had no formal education. He was self–taught in mathematics.

He was the originator of the science of ballistics.In 1535 he discovered how to solve the cubic equationax3 + bx2 + d = 0 (which is a special case of the general cubicequation ax3 + bx2 + cx + d = 0). As was customary at the time, hewould not share the secret of his method with other people.



He had no formal education. He was self–taught in mathematics.He was the originator of the science of ballistics.

In 1535 he discovered how to solve the cubic equationax3 + bx2 + d = 0 (which is a special case of the general cubicequation ax3 + bx2 + cx + d = 0). As was customary at the time, hewould not share the secret of his method with other people.



He had no formal education. He was self–taught in mathematics.He was the originator of the science of ballistics.In 1535 he discovered how to solve the cubic equationax3 + bx2 + d = 0 (which is a special case of the general cubicequation ax3 + bx2 + cx + d = 0). As was customary at the time, hewould not share the secret of his method with other people.


Scipione del Ferro (1465–1526)

He discovered how to solve the cubic equation ax3 + cx + d = 0 butkept his method a secret until, on his death bed, he revealed the secretto his student Antonio Fior.

Armed with this new secret information, Fior challenged Tartaglia to analgebra contest (for money). Each contestant was to create 30problems for the other to solve and whoever solved the most problemswon the money.A few days before the contest, Tartaglia discovered how to solve thegeneral cubic equation ax3 + bx2 + cx + d = 0 and he thus won thecontest easily. (Tartaglia solved all 30 problems that were given to himand Fior solved none.)



He discovered how to solve the cubic equation ax3 + cx + d = 0 butkept his method a secret until, on his death bed, he revealed the secretto his student Antonio Fior.Armed with this new secret information, Fior challenged Tartaglia to analgebra contest (for money). Each contestant was to create 30problems for the other to solve and whoever solved the most problemswon the money.

A few days before the contest, Tartaglia discovered how to solve thegeneral cubic equation ax3 + bx2 + cx + d = 0 and he thus won thecontest easily. (Tartaglia solved all 30 problems that were given to himand Fior solved none.)



He discovered how to solve the cubic equation ax3 + cx + d = 0 butkept his method a secret until, on his death bed, he revealed the secretto his student Antonio Fior.Armed with this new secret information, Fior challenged Tartaglia to analgebra contest (for money). Each contestant was to create 30problems for the other to solve and whoever solved the most problemswon the money.A few days before the contest, Tartaglia discovered how to solve thegeneral cubic equation ax3 + bx2 + cx + d = 0 and he thus won thecontest easily. (Tartaglia solved all 30 problems that were given to himand Fior solved none.)


Cardano and Tartaglia

Cardano badly wanted to learn Tartaglia’s method for solving cubicequations. He persuaded Tartaglia to reveal his secret in exchange forpromising to help Tartaglia get a position as artillery advisor to theSpanish army. Tartaglia taught his method to Cardano but swore him tosecrecy. Later Tartaglia was horrified to learn that Cardano had publishedthe method in Ars Magna. Although Cardano gave full credit to Tartagliafor this major discovery, Tartaglia never forgave Cardano for breaking hispromise. Due to the feud that then started between the two, Tartaglia wasforced to live out the rest of his life in obscurity. In an unrelated matter,Cardano was arrested for heresy for publishing a horoscope of the life ofChrist. He was jailed for several months, lost his university position andwas forbidden from publishing any more books.


Ludovico Ferrari (1522-1565)

He was the personal assistant of Cardano and was taught mathematicsby Cardano.

In 1540, he discovered how to solve the quartic equationax4 + bx3 + cx2 + dx + e = 0 given that the cubic equation could besolved. Once the cubic equation had been solved (by Tartaglia),Cardano was also able to include the method for solving the quarticequation in Ars Magna which was published in 1545.Now that the methods were known for solving polynomial equations ofdegree 1, 2, 3 and 4, the next problem to be tackled was how to solveequation of degree higher than 4. This problem was studied by all ofthe greatest mathematical minds during the next 300 years but nosolution was found.



He was the personal assistant of Cardano and was taught mathematicsby Cardano.In 1540, he discovered how to solve the quartic equationax4 + bx3 + cx2 + dx + e = 0 given that the cubic equation could besolved. Once the cubic equation had been solved (by Tartaglia),Cardano was also able to include the method for solving the quarticequation in Ars Magna which was published in 1545.

Now that the methods were known for solving polynomial equations ofdegree 1, 2, 3 and 4, the next problem to be tackled was how to solveequation of degree higher than 4. This problem was studied by all ofthe greatest mathematical minds during the next 300 years but nosolution was found.



He was the personal assistant of Cardano and was taught mathematicsby Cardano.In 1540, he discovered how to solve the quartic equationax4 + bx3 + cx2 + dx + e = 0 given that the cubic equation could besolved. Once the cubic equation had been solved (by Tartaglia),Cardano was also able to include the method for solving the quarticequation in Ars Magna which was published in 1545.Now that the methods were known for solving polynomial equations ofdegree 1, 2, 3 and 4, the next problem to be tackled was how to solveequation of degree higher than 4. This problem was studied by all ofthe greatest mathematical minds during the next 300 years but nosolution was found.


Neils Henrik Abel (1802–1829)

Abel lived his life in poverty and was not well–recognized during hisshort lifetime. He died of tuberculosis at the age of 26.

Abel proved that there is no method for solving the quintic equationax5 + bx4 + cx3 + dx2 + ex + f = 0 or for solving polynomialequations of any degree higher than 4! Specifically he proved that noformula involving basic arithmetic operations of addition, subtraction,multiplication, division and extraction of roots exists for solvingpolynomial equations of degree higher than 4.

As a tool needed to prove his result on the unsolvability of quinticequations (and equations of higher degree), Abel invented the subjectof Group Theory (which was also invented independently by Galois).The introduction of Group Theory led algebra in a completely newdirection with broader goals beyond just solving equations. This wasthe beginning of the study of what is today called modern algebra orabstract algebra.


The Course of Abstract Algebra after Abel

Inspired by the great discoveries of Abel and Galois regarding the solutionsof polynomial equations and the corresponding discovery of group theory,mathematicians began to view algebra in a different way and began to askdifferent questions. The central problems of algebra were no longer justquestions about how to solve equations. The word “algebra” came to beused more often as a noun to denote different kinds of algebraicstructures. For example, a group is a certain kind of algebra that is definedby certain axioms. However there are many different algebras that aredefined by different axioms. The study of such algebras gradually arose asthey were encountered in trying to solve applied problems. We willconsider a few example of such algebras that you have already encounteredin your previous mathematics courses.


The Algebra of Matrices

A matrix is a rectangular array of numbers such as

A =

−4 7 8 10 −6−8 −5 7 6 −60 5 −10 1 1

.In linear algebra courses we define how to add two matrices of the samesize to obtain another matrix (of the same size). In fact, the set of allm× n matrices (for some given m and n) is a group with the operation ofaddition. However, if we restrict our attention only to the set of squarematrices (say 2× 2 matrices), then we know that we can also define a veryuseful multiplication operation on this set with which we multiply twomatrices of the same size and obtain another matrix (of the same size).Thus we have an algebraic structure (or simply an “algebra”) that consistsof a set with two operations. One of the operations (addition) iscommutative and the other (multiplication) is not. Furthermore not everysquare matrix has a multiplicative inverse so the set of square matrices isnot a group under multiplication.


The Algebra of Vectors

A vector in R3 is an ordered triple of numbers such as v = 〈3,−9, 7〉. Inmultivariable calculus courses we learn that we can define a useful additionoperation on R3. In fact R3 is a group under vector addition. However wealso learn that we can define a useful “multiplication” operation called thecross product. The cross product operation is denoted by the symbol ×.You may recall that the cross product of two vectors in R3 is anothervector in R3. You may also recall that the cross product is notcommutative. This reminds us of the situation that occurs with themultiplication of matrices. For square matrices we generally haveAB 6= BA. Likewise with the cross product we have v×w 6= w× v.However we can be more specific in the case of the cross product becausewe do in fact know that the cross product operation is anti–commutativemeaning that v×w = − (w× v). Thus the the cross product operation ismore “specialized” in some sense than is the operation of matrixmultiplication.


Vector Spaces

In linear algebra courses we are introduced to the algebraic structure calleda vector space. A vector space consists of two sets and two operations.One set, V , is called the set of vectors and its operation is calledaddition. The other set, F , is called the field of scalars. A field is analgebraic structure in its own right which actually has two operations! Inintroductory linear algebra courses we always assume that this field is thefield of real numbers (with the usual addition and multiplication). Besidesthe addition operation, +, on V we define the scalar multiplicationoperation (usually not denoted by any symbol) which is actually anoperation from F × V into V . That is, if k ∈ F and v ∈ V , then kv ∈ V .There is a list of axioms that define exactly what we mean by a vectorspace. For example we require that if k ∈ F and v and w ∈ V , then

k (v + w) = kv + kw.


The General Idea

There are obviously limitless possibilities to the kinds of algebras that wecould study. In general we begin with

A set of sets {S1,S2, . . . ,Sn}A set of operations – some of which may be operations on a single ofthe sets Si and some which may act between the sets

A set of axioms that tell us the rules which the operations mustsatisfy.

This is the “modern” or “abstract” approach to algebra.


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