The set ai is an ideal of a hence it is generated by

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Unformatted text preview: ments x = x0 , x1 , x2 , . . . where xi+1 = xi /pi for some prime pi . Then the sequence of ideals ai = xi A is increasing. The set  ai is an ideal of A, hence it is generated by one element, say y . Then y lies in some aN and this means that the sequence must terminate at aN , i.e. xn is a unit. So a finite factorization exists. 2. Section 1.2 Solving Pythagoras’ equation geometrically. Write the equation as X 2 + Y 2 = 1, where X = x/z , Y = y /z . It is sufficient to find all rational solutions of this equation. Now we know one point on this circle, for example P0 = (−1, 0). For any other point with rational co ordinates, it’s clear that the slope of the line joining it to P0 must be rational (the converse is also not to o hard to see). So write Y X = −1 + m and plug into the equation to get �2 � Y +Y2 = 1 −1 + m which leads to the solution Y = 2m/(m2 + 1), X = (1 − m2 )/(1 + m2 ). Section 1.3 is the Chinese remainder theorem for general rings. 3. gp/Pari example Example. G = bnfclassunit(x^2+5) This G contains lots of arithmetic information. For instance G[2, 1] = [0, 1] gives the number of real and compl√ embeddings of the number field. G[5, 1][1] ex is the class number of the field Q( −5) which is 2. G[5, 2] gives the structure of the class group in terms of its elementary divisors. Here it has to be Z/2. Finally, G[5, 1][3] gives the generators of the cyclic components. Here w√ get a e matrix with colums [2, 0] and [1, 1] which means that the ideal is (2, 1 + 5)....
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This note was uploaded on 01/23/2014 for the course MATH 18.786 taught by Professor Abhinavkumar during the Spring '10 term at MIT.

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