Basic Algebra I: Second Edition by Nathan Jacobson

By Nathan Jacobson

A vintage textual content and conventional reference for a new release, this quantity and its significant other are the paintings of a professional algebraist who taught at Yale for 2 many years. Nathan Jacobson's books own a conceptual and theoretical orientation, and also to their price as lecture room texts, they function beneficial references.

Volume I explores all the themes in general coated in undergraduate classes, together with the rudiments of set conception, workforce idea, earrings, modules, Galois thought, polynomials, linear algebra, and associative algebra. Its complete therapy extends to such rigorous subject matters as Lie and Jordan algebras, lattices, and Boolean algebras. workouts seem through the textual content, in addition to insightful, conscientiously defined proofs. Volume II includes all matters primary to a first-year graduate path in algebra, and it revisits many subject matters from quantity I with higher intensity and class.

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We remark finally that is the only map which can be defined from S/Eα to T to make (8) a commutative diagram. Let β:S/Eα → T satisfy βv = α. Then β( ) = β(v(a) = α(a). Hence β = , by the definition (6). There is a useful generalization of these simple considerations. Suppose we are given a map α:S → T and an equivalence relation E on S. We shall say that α is compatible with E if aEb for a, b in S implies α(a) = α(b). In this case we can define a map of = S/E to T by : ≡ E → α(a). Clearly this is well defined, and if v denotes the natural surjection a → , then α = v, that is, we have the commutativity of 46 In this case the induced map need not be injective.

Let M be the set of natural numbers m such that m ≤ s for every s ∈ S. Then 0 ∈ M, and if s ∈ S then s+ M. Hence 53 M ≠ and so, by the axiom of induction, there exists a natural number l ∈ M such that l+ M. Then l is the required number, since l ≤ s for every s ∈ S. Moreover, l ∈ S since otherwise l < s for every s ∈ S and then l+ ≤ s for every s ∈ S. This contradicts l+ M. The well-ordering property is the basis of the following second principle of induction. Suppose that for every n ∈ we have a statement E(n).

Show that if k and m are positive integers and m ≠ nk for n ∈ then m1/k is irrational. 7 A WORD ON CARDINAL NUMBERS We shall have occasion frequently in this book to use the concept of the cardinal number of a set. At this point it will be well to list the main facts on cardinal numbers that will be required. No proofs will be given. These can be found in a number of places, in particular, in Halmos’ Naive Set Theory. 63 We begin by saying that two sets have the same cardinal number or cardinality (or, are equipotent or just plain equivalent) if there exists a 1–1 (read “one to one”) correspondence between them.

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