Abstract Algebra Theory and Applications - Computer Science ...

1The IntegersThe integers are the building blocks of mathematics. In this chapter wewill investigate the fundamental properties of the integers, including mathematicalinduction, the division algorithm, and the Fundamental Theoremof Arithmetic.1.1 Mathematical InductionSuppose we wish to show that1 + 2 + · · · + n =n(n + 1)2for any natural number n. This formula is easily verified for small numberssuch as n = 1, 2, 3, or 4, but it is impossible to verify for all natural numberson a case-by-case basis. To prove the formula true in general, a more genericmethod is required.Suppose we have verified the equation for the first n cases. We willattempt to show that we can generate the formula for the (n + 1)th casefrom this knowledge. The formula is true for n = 1 since1 =1(1 + 1).2If we have verified the first n cases, thenn(n + 1)1 + 2 + · · · + n + (n + 1) = + n + 12= n2 + 3n + 22(n + 1)[(n + 1) + 1]= .222