By Lindsay N. Childs

This e-book is an off-the-cuff and readable creation to raised algebra on the post-calculus point. The strategies of ring and box are brought via examine of the regular examples of the integers and polynomials. a powerful emphasis on congruence sessions leads in a normal approach to finite teams and finite fields. the hot examples and concept are in-built a well-motivated model and made appropriate by way of many purposes - to cryptography, errors correction, integration, and particularly to easy and computational quantity conception. The later chapters comprise expositions of Rabin's probabilistic primality attempt, quadratic reciprocity, the class of finite fields, and factoring polynomials over the integers. Over a thousand workouts, starting from regimen examples to extensions of idea, are came across through the publication; tricks and solutions for lots of of them are incorporated in an appendix.

The new version comprises issues reminiscent of Luhn's formulation, Karatsuba multiplication, quotient teams and homomorphisms, Blum-Blum-Shub pseudorandom numbers, root bounds for polynomials, Montgomery multiplication, and more.

"At each degree, a large choice of functions is presented...The trouble-free exposition is suitable for the meant audience"

- T.W. Hungerford, Mathematical Reviews

"The kind is leisurely and casual, a guided journey in the course of the foothills, the consultant not able to withstand a number of aspect paths and go back visits to favourite spots..."

- Michael Rosen, American Mathematical Monthly

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Extra info for A Concrete Introduction to Higher Algebra

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By induction applied to S0 , there are c(n − 1, r − 1) of these. The second pile consists exactly of the r-element subsets of S0 , of which there are c(n − 1, r), again by induction. Thus the number of r-element subsets of S is c(n − 1, r − 1) + c(n − 1, r) = c(n, r), which is what we wished to show. The entries of Pascal’s triangle can be computed by the following: Lemma 12. c(n, r) = n! ) Proof. Induction on n. The case n = 0 is obvious: 0! 0! Given n > 0, assume that for all r with 0 ≤ r ≤ n − 1, c(n − 1, r) = Now c(n, 0) = 1 = n!

Thus any two solutions of ax + by = c differ by a solution of ax + by = 0. Conversely, if az + bw = 0 ax + by = c then a(x + z) + b(y + w) = c. In short, Proposition 11. Let x0 , y0 be a solution of ax + by = c. Then the general solution of ax + by = c is of the form x = x0 + z, y = y0 + w, where z, w is any solution of ax + by = 0. This proposition is similar to comparable results in linear algebra and in differential equations: to find the general solution of a non-homogeneous equation (like ax + by = c), find some particular solution to the non-homogeneous equation and add to it the general solution of the corresponding homogeneous equation (like ax + by = 0).

Thus Euclid proves the result by what might be called “infinite descent”: there is no infinite descending chain of natural numbers. The principle of infinite descent can be expressed more affirmatively as the Theorem 8 (Well-Ordering Principle). Any nonempty set of natural numbers has a least element. We can rephrase Euclid’s proof in terms of the well-ordering principle. For any number A > 2, let S be the set of numbers ≥2 which divide A. Since A is a positive divisor of itself, S is nonempty.

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