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.