Algebraic Number Theory Summary

Monday - Friday, June 29 - July 3, 2009

The purpose of the Algebraic Number Theory Working Group is to expose secondary teachers to higher level mathematics topics. During the first hour, the Number Theory Working Group participants attend a lecture together with the Undergraduate Faculty Staff and other participants from the Undergraduate Program. The second hour of the Number Theory Working Group is spent debriefing, discussing, and problem solving with Brian Hopkins facilitating the discussion.

The first lecture provided the participants with an overview of the major encompassing topics that would be developed throughout the three weeks of PCMI and determining everyone's level of comfort with some ideas from algebraic structures that would be used in the remaining sessions.

Some of the items presented in Week 1 were the following:

  1. Which integers are sums/differences of two squares?
  2. How to factor integers in the commutative ring Z[i] = {a + bi | a, b are integers}?
  3. The idea of units, zero devisors, and prime vs. irreducible in the ring Z[i].
  4. The same ideas were then considered in the quotient ring Z/p, where p is a prime.
  5. If p is an odd prime, then p = x2 + y2 iff -1 is a square mod, iff p is equivalent to 1 mod 4, iff p can be factored in Z[i].
  6. When is a prime equal to x2 + 2y2 for some integers x and y?
  7. Is it true that p = x2 + dy2 iff d is a square mod p?
  8. Let a, b be integers and let d = gcd(a,b) then there exists integers x and y such that d is a linear combination of a and b.
  9. Let a, b be integers and d = gcd(a,b), then {ax + by with integers x,y} = {dz such that z is an integer}
  10. If a and b are integers, gcd(a,b) =1, and a|bc then a|c.
  11. If p is an irreducible integer and p|ab then p|a or p|b.
  12. How to generalize the idea of gcd in the set of Gaussian integers Z[i], and in Z[√-d]?
  13. Let A, B be elements of Z[i] and let d be a common devisor of largest norm. Then there exists x, y elements of Z[i] such that d = Ax + By.
  14. Another way of dividing in Z[i]: If A,B are elements of the Gaussian integers, then there exists q, r such that A = qB + r with Norm(r) < Norm(B) non-zero and A/B = q + r/B
  15. Proof that R = {(a + b√-3)/2 such that a,b are integers and a = b(mod2)} is a ring.
  16. For A is an element of the ring R, N(A) = 1 iff A is a unit in R.

The Number Theory Working group decided to meet on Sunday, 05 Jul 2009 at 7:30 pm to work on Problem Set 1.

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This material is based upon work supported by the National Science Foundation under DMS-0940733 and DMS-1441467. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.