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

By Jeffrey Bergen

A Concrete method of summary Algebra offers a great and hugely available advent to summary algebra by means of supplying info at the construction blocks of summary algebra.

It starts off with a concrete and thorough exam of general items akin to integers, rational numbers, genuine numbers, complicated numbers, advanced conjugation, and polynomials. the writer then builds upon those normal items and makes use of them to introduce and inspire complicated options in algebra in a way that's more straightforward to appreciate for many scholars. routines offer a balanced mix of trouble degrees, whereas the volume permits the trainer a range of decisions. the ultimate 4 chapters current the extra theoretical fabric wanted for graduate study.

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

  • Presents a extra normal 'rings first' approach to successfully prime the coed into the the summary fabric of the direction via motivating strategies from prior math classes to steer the dialogue of summary algebra
  • Bridges the distance for college students via displaying how many of the recommendations inside an summary algebra path are literally instruments used to resolve tricky, yet recognized difficulties
  • Builds on really general fabric (Integers, polynomials) and strikes onto extra summary issues, whereas supplying a historic process of introducing teams first as automorphisms
  • Exercises supply a balanced mixture of hassle degrees, whereas the volume permits the trainer a range of selections

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Extra resources for A Concrete Approach to Abstract Algebra: From the Integers to the Insolvability of the Quintic

Sample text

But therefore it must also contain 5, and so on. As a result, it certainly looks like T must contain all the positive integers. Unfortunately, the preceding argument is not a complete proof, as the phase “and so on” is not sufficiently clear or rigorous. However, once again the Well Ordering Principle can be used to finish the proof. The new principle that we are obtaining from the Well Ordering Principle is known as Mathematical Induction, and it can be stated several ways. Mathematical Induction—First Version.

2. An odd integer times an odd integer is always even. 3. Every even integer can be written as the sum of two odd integers. 4. Every even integer greater than 2 can be written as the sum of two prime numbers. 5. Every finite group with an odd number of elements is solvable. 6. Every finite group arises as the Galois group of a finite field extension of the rational numbers. Some of the preceding statements are obviously true, and some are obviously false. For some of the others, we may not yet understand what they are talking about.

Com Proof and Intuition 33 However, we will look at a simpler problem. Suppose the plane is split up into different regions using only straight lines. What is the minimum number of colors needed to color any such map? For an example of the type of situation we are dealing with, consider the following picture. Note that by putting the numbers 1 and 2 into the various regions, we can see that we can color this map with only two colors. 1 1 2 2 1 1 2 2 1 1 2 At this point, you should draw several pictures and consider cases that have 4, 5, or 6 lines dividing the plane and see how many colors you need in each case.

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