Math 772
Spring 2002
Final Exam
- Let
be an odd prime. Show that the nonzero rational integer 
is a square in 
(the 
adic integers) if and only if 
is even and 
- Let
be any field
such that 
Show that there
is a one to one correspondence between the distinct quadratic extensions
of 
and the nontrivial elements of the group 
- Use problems 1 and 2 to
show that if
is an odd prime, then there are precisely 3 distinct
quadratic extensions of 
(namely 
where 
is a primitive 
root of unity in
).
- Show that the rational
numbers are dense in
- Show that the equation
has no
nontrivial solutions with 
rational.