Algebraic Number Theory Summary

Monday - Friday, July 13 - 17, 2009

This week, the Algebraic Number Theory Working Group continued attending lectures during the first hour and meeting separately during the second hour to discuss class notes, debrief, clarify and solve problems.

Some of the topics discussed during this week were the following:

  1. Every ideal in Z[√-5] factors uniquely as a product of prime ideals and is equal to the product of prime ideals.
  2. The result hold true for rings of integers Q[√d] and is true for rings of integers in any field K/Q.
  3. Some stimulating questions to consider were:
    1. When can we write integers in the form x2 + 5y2?
    2. Is it true that the product of two non-principal ideals will always be principal? (This result was proved to be true during the lecture.)
  4. If I is any ideal in Z[√-5] then I = {x | x is in I} is an ideal an I I is principal.
  5. Two ideals P,Q are said to be equivalent if there exists a,b elements of Z[√-5] such that P(a) = Q(b).
  6. The relation P ~ Q iff there exists a,b such that P(a) = Q(b) is an equivalence relation on the ideals of Z[√-5].
  7. The set of all principal ideals in Z[√-5] is an equivalence class for the relation defined in (6) and are called ideal classes.
  8. There are only two ideal classes in Z[√-5].
  9. If I is any non-zero ideal in Z[√-5] then Z[√-5] / I is finite.
  10. If I is any ideal in Z[√-5] then there exists a, an element in I, such that N(a) < 6 [Z[√-5] / I]
  11. If C is any ideal class in Z[√-5], then C contains an ideal such that | Z[√-5] / I | <= 6.
  12. If I is contained in Z[√-5], then the norm of I, | I | is | Z[√-5] / I |.
  13. If I and J are ideals of Z[√-5], then | IJ | = | I | | J |.
  14. If a is an element of Z[√-5], then | a | = | Z[√-5] /(a) | = N(a) = a2 + 5b2.
  15. Suppose I and J are ideals of Z[√-5], and I is contained in J, then there exists an ideal K such that JK = I.
  16. If C is an ideal class for Z[√-5] then there exists an ideal I contained in C such that | I | <= 6.
  17. If I is any ideal of Z[√-5] , then I2 is principal.
  18. If -5 is a square mod p and q, but neither p nor q is equal to x2 + 5y2, then there exists x and y such that pq = x2 + 5y2.
  19. If I and J are not prime, then IJ is prime.
  20. If d < 0, d = 2, 3 (mod 4) then every C class contains ideal I such that I <= (4/pi) * √| d | . (Mikeuski)
  21. In Z[√-5]
    1. if P and Q are principal, then PQ is principal
    2. if P and Q are not principal, then PQ is principal
    3. if P is principal and Q is not principal, then PQ is not principal.
  22. Let R be an integral domain. Suppose A, B ,C, and D are ideals of R.
    If A ~ B and C ~ D, then AC ~ BD.
    In particular, multiplication of ideals gives rise to a multiplication in ideal classes.
  23. If R is the ring of integers in Q(√d) (in any number field) then multiplication of ideals makes the set of ideal classes R into an Abelean Group.
  24. Definition of ideal classes.
    If C and D are idea classes, then we define CD to be the set of classes of cd for any choices of c in C and d in D.
  25. The set of principal ideals is the identity element for multiplication.
  26. Which numbers are of the form a2 + 11b2?
  27. If class number of Rd is 2 then you can differentiate between "good" and "bad" primes (for which d is a square) by a simple congruence modulo some number N.
  28. More generally, there is a polynomial f(x) (with degree = class number) such that a prime p with d a square mod p is "good" iff f(x) has a root mod p.
  29. Q(√d) is thought to have infinitely many d which have class number 1. However, no such proof exists yet.

At our Tuesday 2pm meeting, the working group met with several members of the Undergraduate Faculty Program. Discussion topics included what sort of number theory course might best serve future teachers, the relative merits of traditional elementary number theory content versus algebraic number theory, differences in high school and post-secondary teaching styles and the resulting transition for students, and what training college faculty would like high school graduates to have.

On Friday, 17 July 2009, during the working group presentations, Lori gave a discourse on the guiding questions and an explanation of the mathematics that our working group had dealt with over the past three weeks, to the rest of the SSTP participants. This presentation was followed by Blue giving a pedagogical reflection on our working groups experience from our working group.

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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.