1. Problem 5.4 .1 Show that +, − and × of numbers in Q[√n] are them- selves in [√n].
2. Problem 5.4.2 Show that
1
x+y√n
for x, y ∈ Q (n ot both zero) is of the
form x′ + y′√n for x′, y′ ∈ Q. Deduce that Q[√n] is closed under ÷ by
nonzero members.
3. Problem 5.4.3 If (x1, y1) satisfies x2 − ny2 = k1 and (x2 , y2 ) satisfies x2 − ny2 = k2 , show that (x1x2 + ny1 y2 , x1y2 + x2y1) satisfies x2 − ny2 = k1 k2.
4. Problem 5.4.4 Find a nontrivial solution of x2 −17y2 = −1 by inspection and use it to find a nontrivial solution to x2 − 17y2 = 1.
5. Problem 5.4.5 Similarly find a nontrivial solution of x2 − 37y2 = 1.
6. Using a calculator, if needed, find integers a and b such that 1 ≤ b ≤ 8 and |a − bπ| < 1 . (π is the area of the unit circle.)
8
7. Find four rational numbers p
with |√3 − p | ≤ 1 .
q q q2
8. Find four rational numbers p
with |√5 − p | ≤ 1 .
q q q2
9. Show that if α = a
b
is a rational number, then there are only finitely
many rational numbers p
such that | p − a | < 1 .
q q b q2
10. Find a solution to the Pell equation x2 − 10y2 = 1.
Homework 6, due Monday, March 20
1. Problem 5.4 .1 Show that +, − and × of numbers in Q[√n] are them- selves in [√n].
2. Problem 5.4.2 Show that
1
x+y√n
for x, y ∈ Q (n ot both zero) is of the
form x′ + y′√n for x′, y′ ∈ Q. Deduce that Q[√n] is closed under ÷ by
nonzero members.
3. Problem 5.4.3 If (x1, y1) satisfies x2 − ny2 = k1 and (x2 , y2 ) satisfies x2 − ny2 = k2 , show that (x1x2 + ny1 y2 , x1y2 + x2y1) satisfies x2 − ny2 = k1 k2.
4. Problem 5.4.4 Find a nontrivial solution of x2 −17y2 = −1 by inspection and use it to find a nontrivial solution to x2 − 17y2 = 1.
5. Problem 5.4.5 Similarly find a nontrivial solution of x2 − 37y2 = 1.
6. Using a calculator, if needed, find integers a and b such that 1 ≤ b ≤ 8 and |a − bπ| < 1 . (π is the area of the unit circle.)
8
7. Find four rational numbers p
with |√3 − p | ≤ 1 .
q q q2
8. Find four rational numbers p
with |√5 − p | ≤ 1 .
q q q2
9. Show that if α = a
b
is a rational number, then there are only finitely
many rational numbers p
such that | p − a | < 1 .
q q b q2
10. Find a solution to the Pell equation x2 − 10y2 = 1.
