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A 'unit' is a number in a number system that can divide others, and has all numbers as its set of multiples. There can be multiple units in a system. Also, an easy way to prove a number as unit, is to show that $1$ is a multiple of that.

My questions are below with answers when am able to, and then again for vetting :

Q.1. Number $2$ is not a unit when considered as an element of integers, but is a unit when considered as an element of rational numbers.

=> $2$ cannot have a value as 1/2 in $Z$, but definitely in $R$.

Q.2. In $Z[\sqrt2]$ the numbers like $\sqrt 2$ and 2 are not units.

Q.3. Show that there are more than a thousand units in $Z[\sqrt2]$.

jiten
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    The question is not about Gaussian integers. – lhf Nov 16 '17 at 20:56
  • @Ihf Can I say Integral domain, at least what I found on googling. I am new to the area, and this topic occurred in the book (Numbers and Symmetry, by: Johnston, Richman) right after Gaussian Integers, without any sort of nomenclature. Please modify title, as you feel fit. Also, if there can be a better intro. book for these topics, please state. – jiten Nov 16 '17 at 22:27

2 Answers2

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Hint for Q3: $1+\sqrt2$ is a unit. If $u$ is a unit, so is $u^n$ for all $n$.

lhf
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Each time that $(x+y\sqrt2)(x-y\sqrt2)=\pm1\iff x^2-2y^2=\pm1$ you have a unit in $Z[\sqrt2]$.

See about Pell-Fermat equation for details. In your particular case all the solutions are given by $$x_n+y_n\sqrt2= (1+\sqrt2)^n$$ For each $n\in\mathbb N$ you have a unit $x_n+y_n\sqrt2$.

Piquito
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