Introduction to Algebra

Introduction to Algebra

Peter J. Cameron

Language: English

Pages: 350

ISBN: 0198569130

Format: PDF / Kindle (mobi) / ePub

Developed to meet the needs of modern students, this Second Edition of the classic algebra text by Peter Cameron covers all the abstract algebra an undergraduate student is likely to need. Starting with an introductory overview of numbers, sets and functions, matrices, polynomials, and modular arithmetic, the text then introduces the most important algebraic structures: groups, rings and fields, and their properties. This is followed by coverage of vector spaces and modules with applications to abelian groups and canonical forms before returning to the construction of the number systems, including the existence of transcendental numbers. The final chapters take the reader further into the theory of groups, rings and fields, coding theory, and Galois theory. With over 300 exercises, and web-based solutions, this is an ideal introductory text for Year 1 and 2 undergraduate students in mathematics.

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is an ideal. So the operations on R/I are indeed well defined. Now the rest of the proof involves verifying the axioms, which is routine. The closure laws need no proof, since we have well-defined operations. For the associative law (A1), we have ((I + x) + (I + y)) + (I + z) = (I + (x + y)) + (I + z) = I + ((x + y) + z) = I + (x + (y + z)) = (I + x) + (I + (y + z)) = (I + x) + ((I + y) + (I + z)). The proofs of (A4), (M1), and (D) are very similar. The zero element is I +0 = I, and the inverse of

to a contradiction, so this assumption must be false; there must be infinitely many primes. The second theorem was proved by Pythagoras (or possibly one of his students). This theorem is surrounded by legend: supposedly Hipparchos, a disciple of Pythagoras, was killed (in a shipwreck) by the gods for revealing the disturbing truth that there are ‘irrational’ numbers. √ 2 is irrational; that is, there is no number x = p/q (where p Theorem 1.2 and q are whole numbers) such that x2 = 2. Proof Again

Not every group is cyclic. In the first place, cyclic groups are necessarily abelian: for g m g n = g m+n = g n g m for all m, n. And not all abelian groups are cyclic. The Klein group V4 (see Example 5 in Section 3.2) is abelian but not cyclic. Indeed, we can recognise cyclic groups as follows: Proposition 3.7 A finite group G of order n is cyclic if and only if it contains an element of order n. Proof If g has order n, then the elements g 0 = 1, g 1 = g, . . . , g n−1 of g are all distinct; since

dependence relation, the coefficient of v must be non-zero (else X would be linearly dependent); so we can use the relation to express v as a linear combination of X. So X spans V , and is a basis. (c) implies (b): If X is a basis, then no element of X is a linear combination of the others, so no proper subset is spanning. Vector spaces 157 (b) implies (c): If X is a minimal spanning set, then no element of X is a linear combination of the others (or it could be dropped without losing the

1)n > m). Let q be the last integer x for which xn ≤ m; that is, nq ≤ m but n(q + 1) > m. (It may be that q = 0.) Put r = m − nq. Then r ≥ 0 but r < n; and m = nq + r. The integers As we have seen, subtraction is not always possible for natural numbers. To get round this, we enlarge the number system to include negative numbers as well as positive numbers and zero, giving the set Z = {. . . , −2, −1, 0, 1, 2, 3, . . .} of integers. Thus, we can add, subtract, and multiply integers. The laws we

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