January 16, 2015
You may use results proved in the class after recalling them and stating them explicitly.
1) Show that the set of Pythagorean primitive triples are in bijection with the set of pairs {± ( m
2 − n
2
) , ± 2 mn } where m, n are positive integers that are relatively prime and
( m, n ) not both odd. (Hint: Use the result that if ( x, y, z ) is a Pythagorean triple, then
( x + yi ) = uα
2
, with u a unit in
Z
[ i ] and α ∈
Z
[ i ] = m + ni.
)
2) Use the above exercise to show that there exist positive relativey prime integers, r, s not both odd, such that r, s and m = r
2
+ s
2 are all squares. Write m = c
0 2
, r = a
0 2
, b
0 2
= s and show that a
0 4
+ b
0 4
= c
0 2
.
(This is the ‘descent step argument’ mentioned in the class).
3) Recall that Fermat’s conjecture states that if x n
+ y n
= z n has a solution in integers, then xyz = 0 .
Show that it is enough to prove this conjecture when n is a prime, in order to prove the general statement.
4) Let A ⊆ B be an extension of commutative rings. Recall that the annihilator of an
A -module M is the set { x ∈ A | xm = 0 } for all m ∈ M.
Prove that this is an ideal in A .
An A -module M is said to be faithful if its annihilator ideal is trivial. Prove that if x in
B is integral over A , then there is a faithful A [ x ]-module R which is finitely generated as an A -module.
5) Prove the transitivity of integral extensions.
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