# 2007-10-24

- Hand in the graded homework.
- Hand out new homework, which is due next Friday.
- Announcements:
- Koopa Koo will talk about cyclotomic fields on Friday, while I'm in DC.
- This afternoon: intro to crypto seminar by Robert Bradshaw at 4:40pm in B027.

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 . This is the field defined by a root of . 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 ):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.
- Some properties of ideals
- Proof of general CRT
- Application to ideal generators:
- Each ideal is generated by at most 2 elements (do prove this).
If m is a maximal ideal then is isomorphic to as an module. (State but do not prove -- refer to the book for proof.)