introduction to algebra feb 2008

[...]... theory Galois Theory 299 299 317 Further reading 334 Index 337 1 Introduction The purpose of this chapter is to introduce you to some of the notation and ideas that make up mathematics Much of this may be familiar to you when you begin the study of abstract algebra But, if it is not, I have tried to provide a friendly introduction Your job is to practice unfamiliar skills until you are fluent If you do not... positive integers n Elementary algebra Abu Ja’far Muhammad ibn Musa al-Khwarizmi (whose name gives us the word ‘algorithm’) wrote an algebra textbook which included much of what is still regarded as elementary algebra today The title of his book was Hisab al-jabr w’al-muqabala The word al-jabr means ‘restoring’, referring to the process of moving a negative quantity to the other side of an equation;... positive integers is equal to the square of their sum Exercise 1.14 When the mathematician Gauss was in primary school, his teacher asked the class to add up all the numbers from 1 to 100 Gauss saw that, if he took the sum S= 1+ 2+ 3 + · · · + 99 + 100, 22 Introduction and wrote it down reversed, S = 100 + 99 + 98 + · · · + 2+ 1, then each pair of numbers in the two sums adds up to 101 So 2S = 100 × 101... yourself A ‘Conjecture’ is a statement which is believed to be true but for which we do not yet have a proof Much of what mathematicians do is working to establish a conjecture (or, since not all conjectures turn out to be true, to refute one) Another important part of our work is to make conjectures based on our experience and intuition, for others to prove or disprove (The great twentiethcentury Hungarian... an equation; the word al-muqabala means ‘comparing’, and refers to subtracting equal quantities from both sides of an equation Both processes are familiar to anyone who has to solve an equation! The word al-jabr has, of course, been incorporated into our language as algebra In this section we briefly revise the techniques of elementary algebra 1.6 Formulae and equations tion of symbols like A formula,... Introduction 19 There are several variations on this principle Perhaps, in place of knowing P(0), we know P(1) Then we can conclude that P(n) holds for all n ≥ 1 A similar statement would hold with 100, or any fixed number, replacing 1 It is important to note that, in a proof by induction, we have two jobs: to prove P(0) (called starting the induction) and to prove that the implication from P(n) to. .. natural number satisfying 1 < m < n, then m has a prime factor • • If n ≤ 1 then the statement is vacuously true If n is prime, then it is a prime factor of itself, and the statement is true 21 Introduction • Suppose that n is composite; then n = ab, where 1 < a, b < n By the induction hypothesis, a has a prime factor p Now p is a prime factor of n, and so again the statement is true We have covered... 1.2 Proofs The real answer to our earlier question ‘What is mathematics?’ is: Mathematics is about proofs A proof is nothing but an argument to convince you of the truth of some assertion Mathematical statements require proofs, which should be completely convincing, though you might have to work to understand the details If, after a lot of effort, you are not convinced by an 4 Introduction Table 1.1 The... already, say p1 , p2 , , pn Multiply them together and add one: let N be this number, so that N = p1 p2 · · · pn +1 If N is prime, take it to be the next prime pn+1 Otherwise, take pn+1 to be the smallest prime which divides N Euclid gives us a guarantee that pn+1 is different from all the primes p1 , , pn Take p1 = 2 Use MAPLE or a calculator to find p2 , p3 , , p8 Experiment with taking... ADG, BEH, CF I, AEI, BF G, CDH, AF H, BDI, CEG It is some labour to verify the axioms, but once this is done then the conclusion of the theorem must hold Indeed, the lines DEF and GHI are both parallel to ABC, and they are parallel to one another Here we seem to be a long way from traditional geometry, and it does not seem so stupid to say that A, B, C, are tables and ABC, DEF, are chairs, and . solu- tions to the remaining exercises, further topics, problems, and links to other sites of interest to algebraists. The address is http://www.maths.qmul.ac.uk/ ˜ pjc /algebra/ Thanks are due to many.