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lecture1 - ALGEBRAIC NUMBER THEORY LECTURE 1 SUPPLEMENTARY...

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ALGEBRAIC NUMBER THEORY LECTURE 1 SUPPLEMENTARY NOTES Material covered: Sections 1.1 through 1.3 of textbook. 1. Section 1.1 Recall that to an integral domain A we can associate its field of fractions K = Frac( A ) = { a b : b = negationslash 0 } . More formally, K = { ( a,b ) : a,b A,b = negationslash 0 } c / , where is the equivalence relation ( a,b ) ( c,d ) iff ad bc = 0 (i.e. a = ”). b d A general fractional ideal f of A is a subset of K = Frac( A ) such that (1) af f a A,f f (2) f 1 + f 2 f f 1 ,f 2 f (3) c A such that c f is an ideal of A . A non-example of a fractional ideal is the set K : it satisfies the first two properties but not the third one in general: the problem is that we have inverted “too many” elements. Example. Some principal ideal domains (PIDs): (1) Z is a PID (any nonzero ideal is a subgroup, so is generated by a smallest positive element). (2) For a field k , the ring of univariate polynomials k [ X ] is a PID (take a lowest degree element).
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