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

By Jeffrey Bergen

A Concrete method of summary Algebra offers a superb and hugely available advent to summary algebra by way of supplying information at the development blocks of summary algebra.

It starts off with a concrete and thorough exam of regular gadgets equivalent to integers, rational numbers, genuine numbers, advanced numbers, advanced conjugation, and polynomials. the writer then builds upon those universal 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 amount permits the teacher a range of selections. the ultimate 4 chapters current the extra theoretical fabric wanted for graduate study.

This textual content might 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 traditional 'rings first' approach to successfully prime the coed into the the summary fabric of the path by means of motivating ideas from earlier math classes to steer the dialogue of summary algebra
  • Bridges the space for college kids through exhibiting how lots of the ideas inside an summary algebra direction are literally instruments used to resolve tough, yet recognized difficulties
  • Builds on really normal fabric (Integers, polynomials) and strikes onto extra summary issues, whereas delivering a historic procedure of introducing teams first as automorphisms
  • Exercises offer a balanced mix of trouble degrees, whereas the amount permits the teacher a range of decisions

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

Example text

The proof of the next proposition is an example of a proof that combines both the Well Ordering Principle and proof by contradiction. On the plus side, this proof shows the power and usefulness of the Well Ordering Principle. On the negative side, it is an example of a proof where the key ideas are, unfortunately, camouflaged, and the proof adds little to our intuition or understanding. 4. If n is a positive integer, then n is either a whole number or is irrational. com 28 Chapter 2 After reading a mathematical statement, one should first get a thorough understanding of what it really means before reading the proof.

In doing so, you will be forced to write in a very formal and rigorous way. The arguments you make must be airtight. But never forget that the proofs you write are actually the by-product of hard work, experimentation, and imagination. It is creativity that drives new discoveries in mathematics. Although the proofs you read and write may be very formal objects, try to look for the creativity and ideas that motivated those proofs. When reading a proof, you will justify statements on a line-by-line basis.

We are to consider the numbers √ √ √ √ √ √ √ √ 1, 2, 3, 4, 5, . . , 99, 100, 101, . . √ √ √ √ Some of them, like 1, 4, 9, 16, . . , are obviously whole numbers. The proposition then asserts that all the other numbers on the list √ √ √ √ √ √ √ √ √ 2, 3, 5, 6, 7, 8, 10, . . , 99, 101, . . must be irrational. √ √ Proof. Let n be a positive integer such that n is not a whole number; we must show that n is irrational. We will proceed using a proof by contradiction. Therefore, we will assume that √ n is a rational number but is still not a whole number, and we will arrive at a contradiction.

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