Homework 3
Arithmetic in \(\mathbb{Z}[\sqrt{d}]\)
Instructions
For each problem, please write a complete argument, not only intermediate calculations. Answers without justification may receive zero credit. Collaboration is encouraged, but you must write your own final solutions independently.
1 Divisibility. Prove that \[ 1+\sqrt{-2}\mid a+b\sqrt{-2} \] in \(\mathbb{Z}[\sqrt{-2}]\) if and only if \(a\equiv b\pmod 3\).
2 Norm and conjugates. Let \(d\in\mathbb{Z}\) and \(m\ge1\) be integers. For any \(\alpha\in\mathbb{Z}[\sqrt d]\), a multiplicative inverse of \(\alpha\) modulo \(m\) is an element \(\beta\in \mathbb{Z}[\sqrt d]\) such that \(m\mid (\alpha\beta-1)\).
(1) Prove that \(\alpha\) has a multiplicative inverse modulo \(m\) if and only if \(\gcd(N(\alpha),m)=1\).
(2) Show that if \(c\in\mathbb{Z}\) satisfies \(cN(\alpha)\equiv1\pmod m\), then \(c\overline{\alpha}\) is a multiplicative inverse of \(\alpha\) modulo \(m\).
(3) In \(\mathbb Z[\sqrt5]\), find a multiplicative inverse of \(3+\sqrt5\) modulo \(7\).
Hint
What does the proof of the division algorithm in \(\mathbb{Z}[i]\) tell us about the quotient and the remainder?4 Prime factorization in \(\mathbb{Z}[\sqrt{-2}]\). Factor \(165\) into primes in \(\mathbb{Z}[\sqrt{-2}]\). In particular, explain why every factor in your factorization is a prime.
5 Nonunique factorization in \(\mathbb{Z}[\sqrt{d}]\). Suppose that \(d\in\mathbb{Z}\) is not a perfect square. We show that \(\mathbb{Z}[\sqrt{d}]\) is not a UFD for infinitely many positive and negative values of \(d\).
(1) Prove that \(2\in\mathbb{Z}[\sqrt{d}]\) is a prime whenever \(d\le-3\).
(2) Suppose that \(d>0\) is divisible by a prime number \(p\equiv5\pmod 8\). Prove that \(2\in\mathbb{Z}[\sqrt{d}]\) is again a prime.
(3) Suppose either that \(d\le -3\) or that \(d>0\) is divisible by a prime number \(p\equiv5\pmod 8\). Show that \(\mathbb{Z}[\sqrt{d}]\) is not a UFD by considering the factorizations \[\begin{align*} d&=2\cdot\frac{d}{2}=\sqrt{d}\cdot\sqrt{d},\qquad\qquad\qquad\qquad\text{when~}2\mid d,\\ 1-d&=2\cdot\frac{1-d}{2}=(1+\sqrt{d})(1-\sqrt{d}),\qquad\text{when~}2\nmid d. \end{align*}\] Consequently, the only negative values of \(d\) for which \(\mathbb{Z}[\sqrt{d}]\) is a UFD are \(d=-1,-2\) discussed in class.
6 Units and Pell’s equations. Use the continued fraction expansion of \(\sqrt{14}\) to find all the units in \(\mathbb{Z}[\sqrt{14}]\). Does any unit provide a solution to \(x^2-14y^2=-1\)?
Hint
Complete the square.8 Pell solutions as rational approximations. Let \(d>1\) be a nonsquare integer.
(1) Let \((x,y)=(a,b)\) be the fundamental solution to Pell’s equation \(x^2-dy^2=1\), and put \(\omega=a+b\sqrt{d}\). Then \[\varepsilon_n=\omega^n=x_n+y_n\sqrt{d},\quad n\in\mathbb{Z},\] with \(\pm1\) attached, give all the solutions. Establish the recurrence relations \[ x_{n+2}=2ax_{n+1}-x_n\quad\text{and}\quad y_{n+2}=2ay_{n+1}-y_n \] for all \(n\in\mathbb{Z}\).Hint
First show that \(\omega^{-1}=\overline{\omega}\). Then express \(x_n\) and \(y_n\) as \[ x_{n}=\frac{\omega^n+\overline{\omega}^{n}}{2}\quad\text{and}\quad y_{n}=\frac{\omega^n-\overline{\omega}^{n}}{2\sqrt{d}} \] for all \(n\in\mathbb{Z}\). Use these to verify \[ x_{n+2}+x_n=2ax_{n+1}\quad\text{and}\quad y_{n+2}+y_n=2ay_{n+1} \] for all \(n\in\mathbb{Z}\).(2) Show that every solution \(x,y>0\) to Pell’s equation \(x^2-dy^2=1\) satisfies \[ 0<\frac{x}{y}-\sqrt d <\frac{1}{2\sqrt d\,y^2}. \] Consequently, since \(x^2-dy^2=1\) has infinitely many positive solutions, there are infinitely many rational numbers \(x/y\), where \(x,y>0\) and \(\gcd(x,y)=1\), whose distance to \(\sqrt{d}\) is less than \(1/(2\sqrt{d}\,y^2)\). In particular, the sequence of these rational numbers, when arranged with increasing denominators, converges to \(\sqrt{d}\).
(3) Based on the preceding parts, find a rational number \(x/y\), where \(x,y>0\) and \(\gcd(x,y)=1\), such that \[ 0<\frac{x}{y}-\sqrt{3}<10^{-6}. \] You may use the fact that \((x,y)=(2,1)\) is the fundamental solution to \(x^2-3y^2=1\).
Optional challenge
9 A Diophantine equation. Find all integer solutions \((x,y)\in\mathbb{Z}^2\) to \(x^2+2=y^3\).