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Why Abstract Algebra? (Outline of Chapter 1 of Pinter’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 translation of this word is “reunion”. This refers to the collecting of like terms to solve equations. The word was first coined by the mathematician Abu Ja’far Muhammad ibn Musa Al-Khwarizmi (790 – 850). Omar Khayyam (1048–1131) defined “algebra” to be “the science of solving equations”. Sean Ellermeyer (Kennesaw State University)Why Abstract Algebra? (Outline of Chapter 1 of Pinter’s Book) August 17, 2015 2 / 15 Linear and Quadratic Equations Procedures (formulas) for solving linear equations ax + b = 0, a 6= 0 and quadratic equations ax 2 + bx + c = 0, a 6= 0 were known since ancient times (for instance, in ancient Greek society) but no such procedure was known for solving cubic equations ax 3 + bx 2 + cx + d = 0, a 6= 0 or polynomial equations of any degree n > 3. To find such procedures was a problem that occupied mathematicians for many centuries. Sean Ellermeyer (Kennesaw State University)Why Abstract Algebra? (Outline of Chapter 1 of Pinter’s Book) August 17, 2015 3 / 15 Example of a Cubic Equation An example of a cubic equation is 4x 3 + 12x 2 + x + 3 = 0. The solutions of this equation are x = −3, x = 21 i and x = − 21 i (where i is the complex number defined such that i 2 = −1). It is easily checked that these numbers are solutions but how do we find them? Is there a formula (along the same lines as the quadratic formula) that we can use to solve equations of the form ax 3 + bx 2 + cx + d = 0, a 6= 0? Sean Ellermeyer (Kennesaw State University)Why Abstract Algebra? (Outline of Chapter 1 of Pinter’s Book) August 17, 2015 4 / 15 Developments During the Italian Renaissance Girolamo Cardano (1501–1576). To quote from Charles Pinter, Cardano was a physician, an astrologer, a mathematician, a compulsive gambler, a scoundrel and a heretic. He was initially not successful with his medical practice and he and his wife had to seek refuge in a poorhouse. Sean Ellermeyer (Kennesaw State University)Why Abstract Algebra? (Outline of Chapter 1 of Pinter’s Book) August 17, 2015 5 / 15 Developments During the Italian Renaissance Girolamo Cardano (1501–1576). To quote from Charles Pinter, Cardano was a physician, an astrologer, a mathematician, a compulsive gambler, a scoundrel and a heretic. He was initially not successful with his medical practice and he and his wife had to seek refuge in a poorhouse. He was eventually able to land a position as a lecturer of mathematics. Sean Ellermeyer (Kennesaw State University)Why Abstract Algebra? (Outline of Chapter 1 of Pinter’s Book) August 17, 2015 5 / 15 Developments During the Italian Renaissance Girolamo Cardano (1501–1576). To quote from Charles Pinter, Cardano was a physician, an astrologer, a mathematician, a compulsive gambler, a scoundrel and a heretic. He was initially not successful with his medical practice and he and his wife 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 to give a clinical description of typhus fever. Sean Ellermeyer (Kennesaw State University)Why Abstract Algebra? (Outline of Chapter 1 of Pinter’s Book) August 17, 2015 5 / 15 Developments During the Italian Renaissance Girolamo Cardano (1501–1576). To quote from Charles Pinter, Cardano was a physician, an astrologer, a mathematician, a compulsive gambler, a scoundrel and a heretic. He was initially not successful with his medical practice and he and his wife 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 to give a clinical description of typhus fever. Due to his interest in gambling, he took a great interest in probability and became an expert on the subject. He wrote the book Book on Games of Chance. Sean Ellermeyer (Kennesaw State University)Why Abstract Algebra? (Outline of Chapter 1 of Pinter’s Book) August 17, 2015 5 / 15 Developments During the Italian Renaissance Girolamo Cardano (1501–1576). To quote from Charles Pinter, Cardano was a physician, an astrologer, a mathematician, a compulsive gambler, a scoundrel and a heretic. He was initially not successful with his medical practice and he and his wife 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 to give a clinical description of typhus fever. Due to his interest in gambling, he took a great interest in probability and became an expert on the subject. He wrote the book Book on Games of Chance. He also wrote the famous Ars Magna which means “The Great Art”. It was a book that compiled all that was known about algebra at the time. Sean Ellermeyer (Kennesaw State University)Why Abstract Algebra? (Outline of Chapter 1 of Pinter’s Book) August 17, 2015 5 / 15 Niccolò Fontana Tartaglia (1500–1557) He had no formal education. He was self–taught in mathematics. Sean Ellermeyer (Kennesaw State University)Why Abstract Algebra? (Outline of Chapter 1 of Pinter’s Book) August 17, 2015 6 / 15 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. Sean Ellermeyer (Kennesaw State University)Why Abstract Algebra? (Outline of Chapter 1 of Pinter’s Book) August 17, 2015 6 / 15 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 equation ax 3 + bx 2 + d = 0 (which is a special case of the general cubic equation ax 3 + bx 2 + cx + d = 0). As was customary at the time, he would not share the secret of his method with other people. Sean Ellermeyer (Kennesaw State University)Why Abstract Algebra? (Outline of Chapter 1 of Pinter’s Book) August 17, 2015 6 / 15 Scipione del Ferro (1465–1526) He discovered how to solve the cubic equation ax 3 + cx + d = 0 but kept his method a secret until, on his death bed, he revealed the secret to his student Antonio Fior. Sean Ellermeyer (Kennesaw State University)Why Abstract Algebra? (Outline of Chapter 1 of Pinter’s Book) August 17, 2015 7 / 15 Scipione del Ferro (1465–1526) He discovered how to solve the cubic equation ax 3 + cx + d = 0 but kept his method a secret until, on his death bed, he revealed the secret to his student Antonio Fior. Armed with this new secret information, Fior challenged Tartaglia to an algebra contest (for money). Each contestant was to create 30 problems for the other to solve and whoever solved the most problems won the money. Sean Ellermeyer (Kennesaw State University)Why Abstract Algebra? (Outline of Chapter 1 of Pinter’s Book) August 17, 2015 7 / 15 Scipione del Ferro (1465–1526) He discovered how to solve the cubic equation ax 3 + cx + d = 0 but kept his method a secret until, on his death bed, he revealed the secret to his student Antonio Fior. Armed with this new secret information, Fior challenged Tartaglia to an algebra contest (for money). Each contestant was to create 30 problems for the other to solve and whoever solved the most problems won the money. A few days before the contest, Tartaglia discovered how to solve the general cubic equation ax 3 + bx 2 + cx + d = 0 and he thus won the contest easily. (Tartaglia solved all 30 problems that were given to him and Fior solved none.) Sean Ellermeyer (Kennesaw State University)Why Abstract Algebra? (Outline of Chapter 1 of Pinter’s Book) August 17, 2015 7 / 15 Cardano and Tartaglia Cardano badly wanted to learn Tartaglia’s method for solving cubic equations. He persuaded Tartaglia to reveal his secret in exchange for promising to help Tartaglia get a position as artillery advisor to the Spanish army. Tartaglia taught his method to Cardano but swore him to secrecy. Later Tartaglia was horrified to learn that Cardano had published the method in Ars Magna. Although Cardano gave full credit to Tartaglia for this major discovery, Tartaglia never forgave Cardano for breaking his promise. Due to the feud that then started between the two, Tartaglia was forced 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 of Christ. He was jailed for several months, lost his university position and was forbidden from publishing any more books. Sean Ellermeyer (Kennesaw State University)Why Abstract Algebra? (Outline of Chapter 1 of Pinter’s Book) August 17, 2015 8 / 15 Ludovico Ferrari (1522-1565) He was the personal assistant of Cardano and was taught mathematics by Cardano. Sean Ellermeyer (Kennesaw State University)Why Abstract Algebra? (Outline of Chapter 1 of Pinter’s Book) August 17, 2015 9 / 15 Ludovico Ferrari (1522-1565) He was the personal assistant of Cardano and was taught mathematics by Cardano. In 1540, he discovered how to solve the quartic equation ax 4 + bx 3 + cx 2 + dx + e = 0 given that the cubic equation could be solved. Once the cubic equation had been solved (by Tartaglia), Cardano was also able to include the method for solving the quartic equation in Ars Magna which was published in 1545. Sean Ellermeyer (Kennesaw State University)Why Abstract Algebra? (Outline of Chapter 1 of Pinter’s Book) August 17, 2015 9 / 15 Ludovico Ferrari (1522-1565) He was the personal assistant of Cardano and was taught mathematics by Cardano. In 1540, he discovered how to solve the quartic equation ax 4 + bx 3 + cx 2 + dx + e = 0 given that the cubic equation could be solved. Once the cubic equation had been solved (by Tartaglia), Cardano was also able to include the method for solving the quartic equation in Ars Magna which was published in 1545. Now that the methods were known for solving polynomial equations of degree 1, 2, 3 and 4, the next problem to be tackled was how to solve equation of degree higher than 4. This problem was studied by all of the greatest mathematical minds during the next 300 years but no solution was found. Sean Ellermeyer (Kennesaw State University)Why Abstract Algebra? (Outline of Chapter 1 of Pinter’s Book) August 17, 2015 9 / 15 Neils Henrik Abel (1802–1829) Abel lived his life in poverty and was not well–recognized during his short lifetime. He died of tuberculosis at the age of 26. Sean Ellermeyer (Kennesaw State University)Why Abstract Algebra? (Outline of Chapter 1 of Pinter’s Book) August 17, 2015 10 / 15 Neils Henrik Abel (1802–1829) Abel lived his life in poverty and was not well–recognized during his short lifetime. He died of tuberculosis at the age of 26. Abel proved that there is no method for solving the quintic equation ax 5 + bx 4 + cx 3 + dx 2 + ex + f = 0 or for solving polynomial equations of any degree higher than 4! Specifically he proved that no formula involving basic arithmetic operations of addition, subtraction, multiplication, division and extraction of roots exists for solving polynomial equations of degree higher than 4. Sean Ellermeyer (Kennesaw State University)Why Abstract Algebra? (Outline of Chapter 1 of Pinter’s Book) August 17, 2015 10 / 15 Neils Henrik Abel (1802–1829) Abel lived his life in poverty and was not well–recognized during his short lifetime. He died of tuberculosis at the age of 26. Abel proved that there is no method for solving the quintic equation ax 5 + bx 4 + cx 3 + dx 2 + ex + f = 0 or for solving polynomial equations of any degree higher than 4! Specifically he proved that no formula involving basic arithmetic operations of addition, subtraction, multiplication, division and extraction of roots exists for solving polynomial equations of degree higher than 4. As a tool needed to prove his result on the unsolvability of quintic equations (and equations of higher degree), Abel invented the subject of Group Theory (which was also invented independently by Galois). The introduction of Group Theory led algebra in a completely new direction with broader goals beyond just solving equations. This was the beginning of the study of what is today called modern algebra or abstract algebra. Sean Ellermeyer (Kennesaw State University)Why Abstract Algebra? (Outline of Chapter 1 of Pinter’s Book) August 17, 2015 10 / 15 The Course of Abstract Algebra after Abel Inspired by the great discoveries of Abel and Galois regarding the solutions of polynomial equations and the corresponding discovery of group theory, mathematicians began to view algebra in a different way and began to ask different questions. The central problems of algebra were no longer just questions about how to solve equations. The word “algebra” came to be used more often as a noun to denote different kinds of algebraic structures. For example, a group is a certain kind of algebra that is defined by certain axioms. However there are many different algebras that are defined by different axioms. The study of such algebras gradually arose as they were encountered in trying to solve applied problems. We will consider a few example of such algebras that you have already encountered in your previous mathematics courses. Sean Ellermeyer (Kennesaw State University)Why Abstract Algebra? (Outline of Chapter 1 of Pinter’s Book) August 17, 2015 11 / 15 The Algebra of Matrices A matrix is a rectangular array of numbers −4 7 8 −8 −5 7 A= 0 5 −10 such as 10 −6 6 −6 . 1 1 In linear algebra courses we define how to add two matrices of the same size to obtain another matrix (of the same size). In fact, the set of all m × n matrices (for some given m and n) is a group with the operation of addition. However, if we restrict our attention only to the set of square matrices (say 2 × 2 matrices), then we know that we can also define a very useful multiplication operation on this set with which we multiply two matrices of the same size and obtain another matrix (of the same size). Thus we have an algebraic structure (or simply an “algebra”) that consists of a set with two operations. One of the operations (addition) is commutative and the other (multiplication) is not. Furthermore not every square matrix has a multiplicative inverse so the set of square matrices is not a group under multiplication. Sean Ellermeyer (Kennesaw State University)Why Abstract Algebra? (Outline of Chapter 1 of Pinter’s Book) August 17, 2015 12 / 15 The Algebra of Vectors A vector in R 3 is an ordered triple of numbers such as v = h3, −9, 7i. In multivariable calculus courses we learn that we can define a useful addition operation on R 3 . In fact R 3 is a group under vector addition. However we also learn that we can define a useful “multiplication” operation called the cross product. The cross product operation is denoted by the symbol ×. You may recall that the cross product of two vectors in R 3 is another vector in R 3 . You may also recall that the cross product is not commutative. This reminds us of the situation that occurs with the multiplication of matrices. For square matrices we generally have AB 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 because we do in fact know that the cross product operation is anti–commutative meaning that v × w = − (w × v). Thus the the cross product operation is more “specialized” in some sense than is the operation of matrix multiplication. Sean Ellermeyer (Kennesaw State University)Why Abstract Algebra? (Outline of Chapter 1 of Pinter’s Book) August 17, 2015 13 / 15 Vector Spaces In linear algebra courses we are introduced to the algebraic structure called a 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 called addition. The other set, F , is called the field of scalars. A field is an algebraic structure in its own right which actually has two operations! In introductory linear algebra courses we always assume that this field is the field of real numbers (with the usual addition and multiplication). Besides the addition operation, +, on V we define the scalar multiplication operation (usually not denoted by any symbol) which is actually an operation 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 vector space. For example we require that if k ∈ F and v and w ∈ V , then k (v + w) = kv + kw. Sean Ellermeyer (Kennesaw State University)Why Abstract Algebra? (Outline of Chapter 1 of Pinter’s Book) August 17, 2015 14 / 15 The General Idea There are obviously limitless possibilities to the kinds of algebras that we could 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 of the sets Si and some which may act between the sets A set of axioms that tell us the rules which the operations must satisfy. This is the “modern” or “abstract” approach to algebra. Sean Ellermeyer (Kennesaw State University)Why Abstract Algebra? (Outline of Chapter 1 of Pinter’s Book) August 17, 2015 15 / 15