By Norman R Reilly

This upper-level undergraduate textbook presents a contemporary view of algebra with an eye fixed to new functions that experience arisen in recent times. A rigorous creation to easy quantity concept, earrings, fields, polynomial concept, teams, algebraic geometry and elliptic curves prepares scholars for exploring their functional functions regarding storing, securing, retrieving and speaking details within the digital international. it is going to function a textbook for an undergraduate path in algebra with a robust emphasis on purposes. The e-book bargains a quick creation to straight forward quantity thought in addition to a pretty entire dialogue of significant algebraic structures (such as jewelry, fields, and teams) with a view in their use in bar coding, public key cryptosystems, error-correcting codes, counting thoughts, and elliptic key cryptography. this is often the single access point textual content for algebraic structures that incorporates an intensive creation to elliptic curves, an issue that has leaped to prominence because of its value within the answer of Fermat's final Theorem and its incorporation into the speedily increasing functions of elliptic curve cryptography in shrewdpermanent playing cards. desktop technological know-how scholars will get pleasure from the robust emphasis at the concept of polynomials, algebraic geometry and Groebner bases. the combo of a rigorous advent to summary algebra with an intensive insurance of its functions makes this publication really particular.

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3. 3 + 3 · 4 + 3 · 42 + · · · + 3 · 4n−1 = 4n − 1. 4. 12 + 32 + 52 + · · · + (2n − 1)2 = n3 (2n − 1)(2n + 1). 5. 13 + 23 + 33 + · · · + n 3 = 6. 14 + 24 + 34 + · · · + n 4 = n(n+1) 2 1 30 n(n 2 . + 1)(2n + 1)(3n 2 + 3n − 1). 7. Show that the nth Fibonacci number an satisfies an < 2n . 8. Show that for all n ≥ 2, the Fibonacci numbers satisfy: (i) an an+1 − an−1 an = an2 . (ii) an−1 an+1 − an2 = (−1)n . 9. Show that for all n ≥ 1, the Fibonacci numbers satisfy: a12 + a22 + · · · + an2 = an an+1 .

K rk+1 b (v) a yn+1 =− . b xn+1 23. Just for the purposes of this exercise, for all nonzero integers a, b, c define (a, b, c) to be the greatest integer d such that d divides a, b and c. Show that (i) (a, b, c) = ((a, b), c). (ii) ∃ x, y, z ∈ Z with (a, b, c) = xa + yb + zc. 24. Show that any two consecutive Fibonacci numbers are relatively prime. 25. (i) Let a, b ∈ N, x, y ∈ Z be such that ax + by = 1. Show that (yn + a, −xn + b) = 1 for all n ∈ Z. (ii) Show that (4n + 3, 7n + 5) = 1 for all n ∈ Z.

Show that for all n ∈ N, none of the n consecutive integers (n + 1)! + 2, (n + 1)! + 3, · · · , (n + 1)! + n + 1 is prime. 32 Introduction to Applied Algebraic Systems 16. Show that for all n ∈ N, the integer n! + 1 is not divisible by any prime number less than n + 1. Deduce that there must be infinitely many prime numbers. 17. Show that no rational number satisfies the equation x 2 = 2. 5 Relations and Partitions By a relation R on a set A we mean a subset of A × A. If (a, b) ∈ R, then we also write a R b and we say that a is related to b.

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