PMATH 441/641 Algebraic Number Theory, Solutions to Assignment 3
1: Show that Z
1+ 3
2
is not nitely generated as a Z-module and is not a free Z-module.
Solution: Let u = 1+2 3 . We have u 1 = 3 and so u2 u + 1 = 3. Thus u is a root of the polynomial
2
4

PMATH 441/641 Algebraic Number Theory, Assignment 2
Due Mon June 9
1: (a) List all of the elements u = x + iy Z[i] with 0 y x such that x2 + y 2 = 650.
Solution: Note that 650 = 2 52 13 = (1 + i)(1 i)(2 + i)2 (2 i)2 (3 + 2i)(3 2i). The total number of
ele

PMATH 441/641 Algebraic Number Theory, Assignment 3
1: Show that Z
1+ 3
2
Due Fri June 20
is not nitely generated as a Z-module and is not a free Z-module.
2: Let K = Q( 3 2) = SpanQ 1, 3 2 , 3 4 . Show that OK = Z 3 2 = SpanZ = 1, 3 2 , 3 4 .
3: (a)

PMATH 441/641 Algebraic Number Theory, Assignment 2
Due Mon June 9
1: (a) List all of the elements u = x + iy Z[i] with 0 y x such that x2 + y 2 = 650.
(b) Find the number of elements u = x + iy Z[i] with x2 + y 2 = 3 185 000.
(c) Find the number of facto

PMATH 441/641, Solutions to Assignment 1
1: (a) Find an irreducible element in Z[ 6 i] which is not prime.
Solution: We note that 1 + 2 6 i is irreducible since N (1 + 2 6 i) = 25 so if 1 + 2 6 i was reducible then it
would factor into two elements of nor

PMATH 441/641 Algebraic Number Theory, Assignment 1
Due Fri May 23
1: (a) Find an irreducible element in Z[ 6 i] which is not prime.
(b) Find a prime element in Z18 which is not irreducible.
(c) Find an ideal in Z[ 3 i] which is not principal.
(d) Find a