Algebra: Abstract and Concrete (Stressing Symmetry) (2.5 - download pdf or read online
By Frederick M. Goodman
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Additional resources for Algebra: Abstract and Concrete (Stressing Symmetry) (2.5 Edition)
Suppose without loss of generality that jmj jnj. Define sequences jnj > n1 > n2 0 and q1 ; q2 ; : : : by induction, as follows. 6. DIVISIBILITY IN THE INTEGERS This process must stop after no more than n steps with some remainder nrC1 D 0. 10. m; n/: Proof. Write m D n 1 and n D n0 . nk 2 ; nk Ä 0 1/ 1 1 qk Ä Ä 0 1 q 1 1 The matrix Qk D is invertible with inverse Qk D k . nr a; nr b/, where a b is the first row of Q 1 . m; n/ and nr is a common divisor of m and n. 9, nr is the greatest common divisor of m and n.
M; n/. 9. Show that if p is a prime number and a is any nonzero integer, then either p divides a or p and a are relatively prime. 10. Suppose that a and b are relatively prime integers and that x is an integer. Show that if a divides the product bx, then a divides x. Hint: Use the existence of s; t such that sa C t b D 1. 11. Suppose that a and b are relatively prime integers and that x is an integer. Show that if a divides x and b divides x, then ab divides x. 12. Show that if a prime number p divides a product a1 a2 : : : ar of nonzero integers, then p divides one of the factors.
5. mod n/. mod n/. Proof. a C b/ a0 and b b 0 are divisible by n. a is divisible by n. b a0 b 0 / b0/ ■ 40 1. ALGEBRAIC THEMES We denote by Zn the set of residue classes modulo n. The set Zn has a natural algebraic structure which we now describe. Let A and B be elements of Zn , and let a 2 A and b 2 B; we say that a is a representative of the residue class A, and b a representative of the residue class B. The class Œa C b and the class Œab are independent of the choice of representatives. 5.
Algebra: Abstract and Concrete (Stressing Symmetry) (2.5 Edition) by Frederick M. Goodman