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Author: William A. Stein
ant07/plan/2007-10-24
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# 2007-10-24

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 . 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

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 is isomorphic to as an module. (State but do not prove -- refer to the book for proof.)

2013-05-11 18:33