By Chowdhury K.C.
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Extra resources for A first course in theory of numbers
Let p be aprime ideal of o which is relatively prime to the conductor 5 of o, and let Chapter I. Algebraic Integers 48 be the factorization of the polynomial p(X) = p(X) mod p into irreducibles pi (X) = pi (X) mod p over the residue class field alp, with all p; (X) E o[X] monic. , r , are the different prime ideals o f 0 above p. p:. Proof: Writing O1= 0 and 5 = o/p, we have a canonical isomorphism 01p0 z 0'1p0' r a [ x ] /(p(x)). The first isomorphism follows from the relative primality pO 5 = 0 .
For the ring o , the fractional ideals of o , in other words, the finitely generated nonzero o-submodules of the field of fractions K , no longer form a group - unless o happens to be Dedekind. , to those fractional ideals a of o for which there exists a fractional ideal b such that These form an abelian group, for trivial reasons. The inverse of a is still the fractional ideal a-' = ( x E K xa g o } , I In the case where o is a Dedekind domain, the Picard group is of course nothing but the ideal class group C I K .
Algebraic Integers 70 To end this section, we now want to compare a Dedekind domain o to the ring o(X) = f I f ,g (g E 0, g + 0 mod p for p E X I , where X is a set of prime ideals # 0 of o which contains almost all prime ideals of o. 1), the prime ideals # 0 of o ( X ) are given as px = po(X), for p E X, and it is easily checked that o and o(X) have the same localizations o p = o(X)px. We denote by Cl(o), resp. Cl(o(X)), the ideal class groups of o , resp. o(X). 6) Proposition. There is a canonical exact sequence and one has K *lo: S Z.
A first course in theory of numbers by Chowdhury K.C.