# Mathematics 351, Abstract Algebra Fall 2007

Summaries of lectures

These brief summaries of the material covered in each lecture will normally be posted about 5 days after the lecture.

9/5/07:  (a)  The set of integers (denoted Z) satisfies the Well-ordering Axiom:  Every nonempty subset of the nonnegative integers contains a smallest element.

(b)  Using the well-ordering axiom one can prove the Division Algorithm:  Let a,b belong to Z, b > 0.  Then there exist unique integers q and r such that

a = bq + r, 0 ≤ r < b.

(c) If a,b belong to Z, b ≠ 0, we say “b divides a” and write b│a if a = bt for some integer t.  In this case we say that a is a factor of b or that a is a divisor of b.

(d) If a, b belong to Z we say that an integer c is a common divisor of a and b if c divides a and c divides b.

(e) If a, b belong to Z and are not both 0, we say that c is the greatest common divisor of a and b if c is a common divisor of a and b and if whenever d is a common divisor of a and b then c ≥ d.  We frequently denote the greatest common divisor of a and b by (a,b).

(f)  If a, b belong to Z and are not both 0, then (a,b) = ra + sb for some integers r and s and (a,b) is the smallest positive integer that can be expressed in this form.  This is prove by letting S denote the set of all positive integers that can be written in the form ra + sb for some integers r and s.  Then S is a nonempty set of nonnegative integers so it contains a smallest element, say d.  Then every common divisor of a and b divides d (this is the easy part) and d is a common divisor of a and b (use the division algorithm to prove this).

(g)  If a, b belong to Z and are not both 0 and (a,b) = 1, we say that a and b are relatively prime.

(h) If a divides bc and if (a,b) = 1, then a divides c.  (To see this, use (f) to write 1 = ra + sb and then multiply both sides by c to get c = rac + sbc and note that both summands are divisible by a.

(i)  We say that an integer p  is prime if p ≠ 0, p ≠ ±1, and the only factors of  p are ±1 and ±p.  Note that according to this definition there are negative prime numbers (e.g., -5 is prime).  This is unusual – most authors require that prime integers be positive.