Algebraic Number Theory Summary

Monday - Friday, July 6 - 10, 2009

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. Can we make a ring R that contains Z[√-5] with (1 + √-5)/2 being an element in R?
  2. The definition of an algebraic number and an algebraic integer.
  3. A complex number A is an algebraic integer iff its monic minimal polynomial has integer coefficients, where the minimal polynomial of A is the monic rational poly. of lowest degree having A as a root.
  4. If √d is rational, then a + b√d, with rational a, b, and d, is an algebraic integer iff 2a is an integer and a2  b2d also an integer. We looked at a clarifying example and determined that the only algebraic integers in Q(√-5) are the elements of Z[√-5].
  5. Definition of an ideal in a commutative ring R.
    If R is a commutative ring, then an ideal of r is a subset I of R such that the following conditions are true: I is closed under addition in R, for all x in I and all r in R the product rx = I, and I is non-empty.
  6. The only ideals in Z are the sets nZ = {nx, such that x is an integer}
  7. Definition of a principal ideal.
    In a commutative ring R, an ideal I of R is called a principal ideal if there exits an element a of R such that I = {ax with x element of R} = a(R) = (a).
  8. Every ideal of Z is principal.
  9. We looked at the product of ideals and the fact that principal ideals can be written with a single generator.
  10. Definition of a prime ideal for a commutative ring R.
    An ideal I contained in R is called a prime ideal if I is not equal to the ring R itself and for two elements of R a and b with a and b elements of I, then either a is in I or b is in I.
  11. Definition of a maximal ideal.
    An ideal I contained in R is called maximal if I is not the ring R itself and there are no ideals J bigger than I except the ring itself.
  12. Definition of an integral domain.
    A non-zero commutative ring R is called an integral domain if R has no zero divisors. In particular, an ideal I of R is prime iff R/I is an integral domain.
  13. Definition of a field.
    A non-zero commutative ring is called a field if every non-zero element has a multiplicative inverse. In particular, an ideal I of R is maximal iff R/I is a field.
  14. In general, integral domains are not fields and not all prime ideals are maximal.
  15. For a prime p, if p | a2 + 5b2, but p does not divide the product ab, then the ideal I given by I = (p, a + b√-5) = {x = ty (mod p), when t = a/b (mod p)} is maximal.
  16. Let p be a prime integer,
    1. if -5 is not a square mod p, then pZ[√-5] is maximal
    2. if -5 is not a square mod p, but p does not equal 2 or 5, then pZ[√-5] = (p, a + b√-5)(p, a - b√-5) where these two ideals are maximal
    3. if p = 2, then 2Z[√-5] = (2, 1 + √-5)2 and (2, 1 + √-5) is maximal
    4. if p = 5 then 5Z[√-5] = (√-5)2 and (√-5).
  17. If P is any prime ideal in Z[√-5] then P is either 0 or one of the ideals above.
  18. Discussion of homomorphism, isomorphism, and kernel.

Remaining Questions:

  1. Does every ideal in Z[√-5] factor into a product of prime ideals?
  2. Is the factorization unique?

The working group got together for a problem solving session on Tuesday, 07 Jul 2009 at 7:15 pm and the group is also planning to have another problem solving session on Sunday, 12 Jul 2009 at 7:30 pm.

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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.