Read e-book online A Concrete Approach to Abstract Algebra: From the Integers PDF

By Jeffrey Bergen

ISBN-10: 0123749417

ISBN-13: 9780123749413

A Concrete method of summary Algebra provides a great and hugely obtainable creation to summary algebra by means of offering info at the development blocks of summary algebra.

It starts off with a concrete and thorough exam of primary items similar to integers, rational numbers, actual numbers, complicated numbers, advanced conjugation, and polynomials. the writer then builds upon those universal gadgets and makes use of them to introduce and inspire complex thoughts in algebra in a fashion that's more uncomplicated to appreciate for many scholars. workouts supply a balanced combination of trouble degrees, whereas the volume permits the teacher a range of decisions. the ultimate 4 chapters current the extra theoretical fabric wanted for graduate study.

This textual content can be of specific curiosity to academics and destiny lecturers because it hyperlinks summary algebra to many subject matters which come up in classes in algebra, geometry, trigonometry, precalculus, and calculus.

  • Presents a extra usual 'rings first' approach to successfully major the scholar into the the summary fabric of the path by way of motivating recommendations from earlier math classes to steer the dialogue of summary algebra
  • Bridges the distance for college students by means of displaying how many of the recommendations inside an summary algebra path are literally instruments used to resolve tough, yet recognized difficulties
  • Builds on particularly frequent fabric (Integers, polynomials) and strikes onto extra summary subject matters, whereas delivering a old strategy of introducing teams first as automorphisms
  • Exercises supply a balanced mixture of trouble degrees, whereas the amount permits the teacher a range of selections

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Additional info for A Concrete Approach to Abstract Algebra: From the Integers to the Insolvability of the Quintic

Example text

To do that we consider what logical consequences would follow if the statement were indeed false. Suppose that one of the logical consequences of our statement being false is the truth of a new statement that we know cannot possibly be true because it contradicts other statements that we already know to be true. In order for this contradiction to have arisen, somewhere in our mathematical argument there must have been a mistake or faulty piece of logic. However, if we were careful and made no mathematical mistakes or logical errors, then the only thing that could have caused our contradiction was the presumption that our original statement was false.

In reality, the path to a solution is rarely a direct route. But this is not Copyright © 2010 by Elsevier Inc. All rights reserved. 19 20 Chapter 2 what will come across to your reader. They will only see a direct path to the solution. They will not be aware of all the hard work and false starts. Much of the intuition and imagination needed to solve the problem probably never comes across to your reader. The proof your reader will read is likely to be relatively short and extremely formal. At certain points, formality and precision are crucial in the writing of mathematics.

We need to show that no matter how many lines are used to divide up the plane, we can color the plane using only two colors. To this end, we let T = n ∈ N | whenever a plane is divided up by n lines, it can be colored with 2 colors . We will be done if we can show that T contains all positive integers, and we will do this by applying Mathematical Induction. Therefore, we need to show that T satisfies properties (a) and (b). To show that T satisfies property (a), we need to consider the situation where there is only one line cutting across the plane.

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A Concrete Approach to Abstract Algebra: From the Integers to the Insolvability of the Quintic by Jeffrey Bergen

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