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  1. Hand in the graded homework.
  2. Hand out new homework, which is due next Friday.
  3. Announcements:
    1. Koopa Koo will talk about cyclotomic fields on Friday, while I'm in DC.
    2. This afternoon: intro to crypto seminar by Robert Bradshaw at 4:40pm in B027.
  4. Correction: O_K may require arbitrarily many generators, though its ideals require at most 2. General proof requires several ideas that I haven't talked about in class. Here's an example. Consider the degree 5 subfield of the degree 150 field $\mathbf{Q}(\zeta_{151})$. This is the field defined by a root of $f = {x}^{5}  + {x}^{4}  - {60 \cdot {x}^{3} } - {12 \cdot {x}^{2} } + {784 \cdot x} + 128$. In this field, we have that 2 splits as a product of 5 primes. We check this with a computation, though it is also something that can be seen easily as a special case of Class Field Theory (for $\mathbf{Q}$):

    •    sage: f = x^5+x^4-60*x^3-12*x^2+784*x+128
         sage: K.<a> = NumberField(f)
         sage: len(K.factor_integer(2))
  5. State and very quickly prove classical CRT using cardinality argument.
  6. Some properties of ideals
  7. Proof of general CRT
  8. Application to ideal generators:
    1. Each ideal is generated by at most 2 elements (do prove this).
    2. If m is a maximal ideal then $m^n/m^{n+1}$ is isomorphic to $\mathcal{O}_K/m$ as an $\mathcal{O}_K$ module. (State but do not prove -- refer to the book for proof.)

2013-05-11 18:33