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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.
 Spring '10
 AbhinavKumar
 Algebra, Number Theory, Fractions

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